Theorem axc15 2430

Description: Derivation of set.mm's original ax-c15 38845 from ax-c11n 38844 and the shorter ax-12 2178 that has replaced it.

Theorem ax12 2431 shows the reverse derivation of ax-12 2178 from ax-c15 38845.

Normally, axc15 2430 should be used rather than ax-c15 38845, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2380. (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 1969	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		dveeq2 2386	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3		ax12v 2179	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		equeuclr 2022	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
5	4	sps 2186	. . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
6	4	imim1d 82	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) → (𝑥 = 𝑦 → 𝜑)))
7	6	al2imi 1813	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	7	imim2d 57	. . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	5, 8	imim12d 81	. . . 4 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
10	2, 3, 9	syl6mpi 67	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
11	10	exlimdv 1932	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
12	1, 11	mpi 20	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782

This theorem is referenced by:  ax12  2431  ax12b  2432  equs5  2468  ax12vALT  2477  bj-ax12v3ALT  36652