Theorem dveeq2-o 39303

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2383 using ax-c15 39259. (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 1912	. 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2		ax-5 1912	. 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3		equequ2 2028	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
4	1, 2, 3	dvelimf-o 39299	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-c5 39253  ax-c4 39254  ax-c7 39255  ax-c10 39256  ax-c11 39257  ax-c9 39260

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786

This theorem is referenced by:  ax12eq  39311  ax12el  39312  ax12inda  39318  ax12v2-o  39319