Theorem axc15 2459

Description: Derivation of set.mm's original ax-c15 34696 from ax-c11n 34695 and the shorter ax-12 2203 that has replaced it.

Theorem ax12 2460 shows the reverse derivation of ax-12 2203 from ax-c15 34696.

Normally, axc15 2459 should be used rather than ax-c15 34696, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

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 2059	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		dveeq2 2454	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3		ax12v 2204	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		equequ2 2111	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
5	4	sps 2209	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
6		nfa1 2184	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 𝑧 = 𝑦
7	5	imbi1d 330	. . . . . . . 8 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
8	6, 7	albid 2246	. . . . . . 7 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
9	8	imbi2d 329	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10	5, 9	imbi12d 333	. . . . 5 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
11	3, 10	mpbii 223	. . . 4 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
12	2, 11	syl6 35	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
13	12	exlimdv 2013	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
14	1, 13	mpi 20	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1629  ∃wex 1852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858

This theorem is referenced by:  ax12  2460  ax12b  2491  equs5  2497  ax12vALT  2575  bj-ax12v3ALT  33012