Theorem hbsb4 905

Description: A variable not free remains so after substitution with a distinct variable.

Hypothesis

Ref	Expression
hbsb4.1	⊢ (φ → ∀zφ)

Assertion

Ref	Expression
hbsb4	⊢ (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ))

Proof of Theorem hbsb4

Step	Hyp	Ref	Expression
1		ax-8 798	. . . . . . 7 ⊢ (x = z → (x = y → z = y))
2	1	a4s 682	. . . . . 6 ⊢ (∀x x = z → (x = y → z = y))
3	2	eq4s 822	. . . . 5 ⊢ (∀z z = x → (x = y → z = y))
4	3	del35 836	. . . 4 ⊢ (∀z z = x → (∀x x = y → ∀z z = y))
5	4	con3d 87	. . 3 ⊢ (∀z z = x → (¬ ∀z z = y → ¬ ∀x x = y))
6		hbsb2 873	. . . 4 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))
7		ax-10 800	. . . . 5 ⊢ (∀x x = z → (∀x[y / x]φ → ∀z[y / x]φ))
8	7	eq4s 822	. . . 4 ⊢ (∀z z = x → (∀x[y / x]φ → ∀z[y / x]φ))
9	6, 8	syl9r 56	. . 3 ⊢ (∀z z = x → (¬ ∀x x = y → ([y / x]φ → ∀z[y / x]φ)))
10	5, 9	syld 27	. 2 ⊢ (∀z z = x → (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ)))
11		eq5 824	. . . . . 6 ⊢ (∀x x = y → ∀z∀x x = y)
12		ax-4 673	. . . . . . 7 ⊢ (∀x x = y → x = y)
13	12	19.20i 691	. . . . . 6 ⊢ (∀z∀x x = y → ∀z x = y)
14		sbequ2 864	. . . . . . . 8 ⊢ (x = y → ([y / x]φ → φ))
15	14	a4s 682	. . . . . . 7 ⊢ (∀z x = y → ([y / x]φ → φ))
16		sbequ1 863	. . . . . . . . 9 ⊢ (x = y → (φ → [y / x]φ))
17	16	19.20ii 692	. . . . . . . 8 ⊢ (∀z x = y → (∀zφ → ∀z[y / x]φ))
18		hbsb4.1	. . . . . . . 8 ⊢ (φ → ∀zφ)
19	17, 18	syl5 22	. . . . . . 7 ⊢ (∀z x = y → (φ → ∀z[y / x]φ))
20	15, 19	syld 27	. . . . . 6 ⊢ (∀z x = y → ([y / x]φ → ∀z[y / x]φ))
21	11, 13, 20	3syl 21	. . . . 5 ⊢ (∀x x = y → ([y / x]φ → ∀z[y / x]φ))
22	21	a1d 14	. . . 4 ⊢ (∀x x = y → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ([y / x]φ → ∀z[y / x]φ)))
23		sb4 861	. . . . 5 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))
24		eq6 826	. . . . . . . 8 ⊢ (¬ ∀z z = x → ∀x ¬ ∀z z = x)
25		eq6 826	. . . . . . . 8 ⊢ (¬ ∀z z = y → ∀x ¬ ∀z z = y)
26	24, 25	hban 704	. . . . . . 7 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ∀x(¬ ∀z z = x ∧ ¬ ∀z z = y))
27		eq6 826	. . . . . . . . 9 ⊢ (¬ ∀z z = x → ∀z ¬ ∀z z = x)
28		eq6 826	. . . . . . . . 9 ⊢ (¬ ∀z z = y → ∀z ¬ ∀z z = y)
29	27, 28	hban 704	. . . . . . . 8 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ∀z(¬ ∀z z = x ∧ ¬ ∀z z = y))
30		ax-12 802	. . . . . . . . 9 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
31	30	imp 277	. . . . . . . 8 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (x = y → ∀z x = y))
32	18	a1i 7	. . . . . . . 8 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (φ → ∀zφ))
33	29, 31, 32	hbimd 787	. . . . . . 7 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ((x = y → φ) → ∀z(x = y → φ)))
34	26, 33	19.20d 693	. . . . . 6 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (∀x(x = y → φ) → ∀x∀z(x = y → φ)))
35		sb2 859	. . . . . . . 8 ⊢ (∀x(x = y → φ) → [y / x]φ)
36	35	19.20i 691	. . . . . . 7 ⊢ (∀z∀x(x = y → φ) → ∀z[y / x]φ)
37	36	a7s 689	. . . . . 6 ⊢ (∀x∀z(x = y → φ) → ∀z[y / x]φ)
38	34, 37	syl6 23	. . . . 5 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (∀x(x = y → φ) → ∀z[y / x]φ))
39	23, 38	syl9 55	. . . 4 ⊢ (¬ ∀x x = y → ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ([y / x]φ → ∀z[y / x]OFONT COLOR="#0000FF">φ)))
40	22, 39	pm2.61i 110	. . 3 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → ([y / x]φ → ∀z[y / x]φ))
41	40	exp 291	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ)))
42	10, 41	pm2.61i 110	1 ⊢ (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  hbsb4t 906  ddelimf 908  sbco2 913  hbsb 987  sbal1 996  hbab 1096

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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