Theorem ax11v2 1213

Description: Recovery of ax11o 1215 from ax11v 1263 without using ax-11 965. The hypothesis is even weaker than ax11v 1263, with z both distinct from x and not occurring in φ. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1215.

Hypothesis

Ref	Expression
ax11v2.1	⊢ (x = z → (φ → ∀x(x = z → φ)))

Assertion

Ref	Expression
ax11v2	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Distinct variable groups:   x,z   y,z   φ,z

Proof of Theorem ax11v2

Step	Hyp	Ref	Expression
1		a9e 1123	. 2 ⊢ ∃z z = y
2		ax11v2.1	. . . . 5 ⊢ (x = z → (φ → ∀x(x = z → φ)))
3		equequ2 1133	. . . . . . 7 ⊢ (z = y → (x = z ↔ x = y))
4	3	adantl 388	. . . . . 6 ⊢ ((¬ ∀x x = y ⋀ z = y) → (x = z ↔ x = y))
5		dveeq2 1210	. . . . . . . . 9 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))
6	5	imp 350	. . . . . . . 8 ⊢ ((¬ ∀x x = y ⋀ z = y) → ∀x z = y)
7		hba1 1001	. . . . . . . . 9 ⊢ (∀x z = y → ∀x∀x z = y)
8	3	imbi1d 612	. . . . . . . . . 10 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ)))
9	8	a4s 982	. . . . . . . . 9 ⊢ (∀x z = y → ((x = z → φ) ↔ (x = y → φ)))
10	7, 9	albid 1102	. . . . . . . 8 ⊢ (∀x z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
11	6, 10	syl 10	. . . . . . 7 ⊢ ((¬ ∀x x = y ⋀ z = y) → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
12	11	imbi2d 611	. . . . . 6 ⊢ ((¬ ∀x x = y ⋀ z = y) → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ))))
13	4, 12	imbi12d 625	. . . . 5 ⊢ ((¬ ∀x x = y ⋀ z = y) → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ)))))
14	2, 13	mpbii 193	. . . 4 ⊢ ((¬ ∀x x = y ⋀ z = y) → (x = y → (φ → ∀x(x = y → φ))))
15	14	ex 373	. . 3 ⊢ (¬ ∀x x = y → (z = y → (x = y → (φ → ∀x(x = y → φ)))))
16	15	19.23adv 1212	. 2 ⊢ (¬ ∀x x = y → (∃z z = y → (x = y → (φ → ∀x(x = y → φ)))))
17	1, 16	mpi 44	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954  ∃wex 978

This theorem is referenced by:  ax11a2 1214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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