Theorem dveeq2-o 40084

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2407 using ax-c15 40040. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq2-o	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2-o

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1942	. 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2		ax-5 1942	. 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3		equequ2 2058	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
4	1, 2, 3	dvelimf-o 40080	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-c5 40034  ax-c4 40035  ax-c7 40036  ax-c10 40037  ax-c11 40038  ax-c9 40041

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816

This theorem is used by:  ax12eq  40092  ax12el  40093  ax12inda  40099  ax12v2-o  40100