Theorem axc15 2457

Description: Derivation of set.mm's original ax-c15 39704 from ax-c11n 39703 and the shorter ax-12 2216 that has replaced it.

Theorem ax12 2458 shows the reverse derivation of ax-12 2216 from ax-c15 39704.

Normally, axc15 2457 should be used rather than ax-c15 39704, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 26-Mar-2023.) (New usage is discouraged.)

Assertion

Ref	Expression
axc15	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Proof of Theorem axc15

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 2002	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		dveeq2 2413	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3		ax12v 2217	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		equeuclr 2056	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
5	4	sps 2224	. . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
6	4	imim1d 83	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑)))
7	6	al2imi 1848	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	7	imim2d 58	. . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	5, 8	imim12d 82	. . . 4 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
10	2, 3, 9	syl6mpi 68	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
11	10	exlimdv 1966	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
12	1, 11	mpi 21	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  ax12  2458  ax12b  2459  equs5  2495  ax12vALT  2504  bj-ax12v3ALT  37352