Theorem ax11v2 1813

Description: Recovery of ax11o 1815 from ax11v 1820 without using ax-11 1499. The hypothesis is even weaker than ax11v 1820, 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 1815. (Contributed by NM, 2-Feb-2007.)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11v2

Step	Hyp	Ref	Expression
1		a9e 1689	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		ax11v2.1	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
3		equequ2 1706	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
4	3	adantl 275	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
5		dveeq2 1808	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6	5	imp 123	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
7		hba1 1533	. . . . . . . . 9 ⊢ (∀𝑥 𝑧 = 𝑦 → ∀𝑥∀𝑥 𝑧 = 𝑦)
8	3	imbi1d 230	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
9	8	sps 1530	. . . . . . . . 9 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
10	7, 9	albidh 1473	. . . . . . . 8 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
11	6, 10	syl 14	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
12	11	imbi2d 229	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
13	4, 12	imbi12d 233	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
14	2, 13	mpbii 147	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
15	14	ex 114	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
16	15	exlimdv 1812	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
17	1, 16	mpi 15	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  ax11a2  1814