Theorem wl-dveeq12 33441

Description: The current form of ax-13 2282 has a particular disadvantage: The condition ¬ 𝑥 = 𝑦 is less versatile than the general form ¬ ∀𝑥𝑥 = 𝑦. You need ax-10 2059 to arrive at the more general form presented here. You need 19.8a 2090 (or ax-12 2087) to restore 𝑦 = 𝑧 from ∃𝑥𝑦 = 𝑧 again. (Contributed by Wolf Lammen, 9-Jun-2021.)

Assertion

Ref	Expression
wl-dveeq12	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-dveeq12

Step	Hyp	Ref	Expression
1		exnal 1794	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		hbe1 2061	. . 3 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦)
3		ax13lem2 2332	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2284	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
5	3, 4	syldc 48	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6	5	aleximi 1799	. . . 4 ⊢ (∀𝑥∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
7	6	com12 32	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥∃𝑥 𝑧 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
8		hbe1a 2062	. . 3 ⊢ (∃𝑥∀𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
9	2, 7, 8	syl56 36	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
10	1, 9	sylbir 225	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745

This theorem is referenced by: (None)