Theorem dveeq2-o 40193

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 2410 using ax-c15 40149. (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 1945	. 2 ⊢ (𝑧 = 𝑤 → ∀𝑥 𝑧 = 𝑤)
2		ax-5 1945	. 2 ⊢ (𝑧 = 𝑦 → ∀𝑤 𝑧 = 𝑦)
3		equequ2 2061	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
4	1, 2, 3	dvelimf-o 40189	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-c5 40143  ax-c4 40144  ax-c7 40145  ax-c10 40146  ax-c11 40147  ax-c9 40150

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  ax12eq  40201  ax12el  40202  ax12inda  40208  ax12v2-o  40209