Theorem sbcom2 992

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).

Assertion

Ref	Expression
sbcom2	⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ)

Distinct variable group(s):   x,z   x,w   y,z

Proof of Theorem sbcom2

Step	Hyp	Ref	Expression
1		alcom 715	. . . . . 6 ⊢ (∀z∀x(x = y → (z = w → φ)) ↔ ∀x∀z(x = y → (z = w → φ)))
2		bi2.04 141	. . . . . . . . 9 ⊢ ((x = y → (z = w → φ)) ↔ (z = w → (x = y → φ)))
3	2	bial 695	. . . . . . . 8 ⊢ (∀x(x = y → (z = w → φ)) ↔ ∀x(z = w → (x = y → φ)))
4		19.21v 942	. . . . . . . 8 ⊢ (∀x(z = w → (x = y → φ)) ↔ (z = w → ∀x(x = y → φ)))
5	3, 4	bitr 151	. . . . . . 7 ⊢ (∀x(x = y → (z = w → φ)) ↔ (z = w → ∀x(x = y → φ)))
6	5	bial 695	. . . . . 6 ⊢ (∀z∀x(x = y → (z = w → φ)) ↔ ∀z(z = w → ∀x(x = y → φ)))
7		19.21v 942	. . . . . . 7 ⊢ (∀z(x = y → (z = w → φ)) ↔ (x = y → ∀z(z = w → φ)))
8	7	bial 695	. . . . . 6 ⊢ (∀x∀z(x = y → (z = w → φ)) ↔ ∀x(x = y → ∀z(z = w → φ)))
9	1, 6, 8	3bitr3 156	. . . . 5 ⊢ (∀z(z = w → ∀x(x = y → φ)) ↔ ∀x(x = y → ∀z(z = w → φ)))
10	9	a1i 7	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → (∀z(z = w → ∀x(x = y → φ)) ↔ ∀x(x = y → ∀z(z = w → φ))))
11		sb4b 862	. . . . 5 ⊢ (¬ ∀z z = w → ([w / z][y / x]φ ↔ ∀z(z = w → [y / x]φ)))
12		sb4b 862	. . . . . . 7 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))
13	12	imbi2d 464	. . . . . 6 ⊢ (¬ ∀x x = y → ((z = w → [y / x]φ) ↔ (z = w → ∀x(x = y → φ))))
14	13	bialdv 935	. . . . 5 ⊢ (¬ ∀x x = y → (∀z(z = w → [y / x]φ) ↔ ∀z(z = w → ∀x(x = y → φ))))
15	11, 14	sylan9bbr 419	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([w / z][y / x]φ ↔ ∀z(z = w → ∀x(x = y → φ))))
16		sb4b 862	. . . . 5 ⊢ (¬ ∀x x = y → ([y / x][w / z]φ ↔ ∀x(x = y → [w / z]φ)))
17		sb4b 862	. . . . . . 7 ⊢ (¬ ∀z z = w → ([w / z]φ ↔ ∀z(z = w → φ)))
18	17	imbi2d 464	. . . . . 6 ⊢ (¬ ∀z z = w → ((x = y → [w / z]φ) ↔ (x = y → ∀z(z = w → φ))))
19	18	bialdv 935	. . . . 5 ⊢ (¬ ∀z z = w → (∀x(x = y → [w / z]φ) ↔ ∀x(x = y → ∀z(z = w → φ))))
20	16, 19	sylan9bb 418	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([y / x][w / z]φ ↔ ∀x(x = y → ∀z(z = w → φ))))
21	10, 15, 20	3bitr4d 424	. . 3 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
22	21	exp 291	. 2 ⊢ (¬ ∀x x = y → (¬ ∀z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ)))
23		eq5 824	. . . 4 ⊢ (∀x x = y → ∀z∀x x = y)
24		sbequ12 865	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
25	24	a4s 682	. . . 4 ⊢ (∀x x = y → (φ ↔ [y / x]φ))
26	23, 25	bisbd 897	. . 3 ⊢ (∀x x = y → ([w / z]φ ↔ [w / z][y / x]φ))
27		sbequ12 865	. . . 4 ⊢ (x = y → ([w / z]φ ↔ [y / x][w / z]φ))
28	27	a4s 682	. . 3 ⊢ (∀x x = y → ([w / z]φ ↔ [y / x][w / z]φ))
29	26, 28	bitr3d 408	. 2 ⊢ (∀x x = y → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
30		sbequ12 865	. . . 4 ⊢ (z = w → ([y / x]φ ↔ [w / z][y / x]φ))
31	30	a4s 682	. . 3 ⊢ (∀z z = w → ([y / x]φ ↔ [w / z][y / x]φ))
32		eq5 824	. . . 4 ⊢ (∀z z = w → ∀x∀z z = w)
33		sbequ12 865	. . . . 5 ⊢ (z = w → (φ ↔ [w / z]φ))
34	33	a4s 682	. . . 4 ⊢ (∀z z = w → (φ ↔ [w / z]φ))
35	32, 34	bisbd 897	. . 3 ⊢ (∀z z = w → ([y / x]φ ↔ [y / x][w / z]φ))
36	31, 35	bitr3d 408	. 2 ⊢ (∀z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
37	22, 29, 36	pm2.61ii 113	1 ⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797  [wsb 852

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  ax-17 925

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

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