Theorem ddelimf2 907

Description: Proof of ddelimf 908 without using ax-11 801. This may be useful in a study to determine whether ax-11 801 can be derived from the others, which is currently unknown.

Hypotheses

Ref	Expression
ddelimf2.1	⊢ (φ → ∀xφ)
ddelimf2.2	⊢ (ψ → ∀zψ)
ddelimf2.3	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
ddelimf2	⊢ (¬ ∀x x = y → (ψ → ∀xψ))

Proof of Theorem ddelimf2

Step	Hyp	Ref	Expression
1		ax-10 800	. . . . . 6 ⊢ (∀z z = x → (∀z∀z(z = y → φ) → ∀x∀z(z = y → φ)))
2	1	eq4s 822	. . . . 5 ⊢ (∀x x = z → (∀z∀z(z = y → φ) → ∀x∀z(z = y → φ)))
3		hba1 698	. . . . 5 ⊢ (∀z(z = y → φ) → ∀z∀z(z = y → φ))
4	2, 3	syl5 22	. . . 4 ⊢ (∀x x = z → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
5	4	a1d 14	. . 3 ⊢ (∀x x = z → (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ))))
6		eq6 826	. . . . . 6 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
7		eq6 826	. . . . . 6 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
8	6, 7	hban 704	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ∀z(¬ ∀x x = z ∧ ¬ ∀x x = y))
9		eq6 826	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
10		eq6 826	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀x ¬ ∀x x = y)
11	9, 10	hban 704	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ∀x(¬ ∀x x = z ∧ ¬ ∀x x = y))
12		ax-12 802	. . . . . . 7 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (z = y → ∀x z = y)))
13	12	imp 277	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (z = y → ∀x z = y))
14		ddelimf2.1	. . . . . . 7 ⊢ (φ → ∀xφ)
15	14	a1i 7	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (φ → ∀xφ))
16	11, 13, 15	hbimd 787	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → ((z = y → φ) → ∀x(z = y → φ)))
17	8, 16	hbald 790	. . . 4 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
18	17	exp 291	. . 3 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ))))
19	5, 18	pm2.61i 110	. 2 ⊢ (¬ ∀x x = y → (∀z(z = y → φ) → ∀x∀z(z = y → φ)))
20		ddelimf2.2	. . 3 ⊢ (ψ → ∀zψ)
21		ddelimf2.3	. . 3 ⊢ (z = y → (φ ↔ ψ))
22	20, 21	eqsal 833	. 2 ⊢ (∀z(z = y → φ) ↔ ψ)
23	22	bial 695	. 2 ⊢ (∀x∀z(z = y → φ) ↔ ∀xψ)
24	19, 22, 23	3imtr3g 425	1 ⊢ (¬ ∀x x = y → (ψ → ∀xψ))

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

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-12 802

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

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