Theorem dveeq2-o 39688

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2410 using ax-c15 39644. (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 1940	. 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2		ax-5 1940	. 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3		equequ2 2056	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
4	1, 2, 3	dvelimf-o 39684	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-c5 39638  ax-c4 39639  ax-c7 39640  ax-c10 39641  ax-c11 39642  ax-c9 39645

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is referenced by:  ax12eq  39696  ax12el  39697  ax12inda  39703  ax12v2-o  39704