Theorem ax12v2OLD 2342

Description: Obsolete proof of ax12v 2048 as of 24-Mar-2021. (Contributed by NM, 12-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ax12v2OLD.1	⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))

Assertion

Ref	Expression
ax12v2OLD	⊢ (¬ ∀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 ax12v2OLD

Step	Hyp	Ref	Expression
1		ax6ev 1890	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		dveeq2 2298	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3		ax12v2OLD.1	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		equequ2 1953	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
5	4	sps 2055	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
6		nfa1 2028	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 𝑧 = 𝑦
7	5	imbi1d 331	. . . . . . . 8 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
8	6, 7	albid 2090	. . . . . . 7 ⊢ (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
9	8	imbi2d 330	. . . . . 6 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10	5, 9	imbi12d 334	. . . . 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 1861	. 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 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  ax12a2OLD  2343