Theorem dveeq1-o 39571

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2414 using ax-c11 . (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq1-o	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-5 1933	. 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2		ax-5 1933	. 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3		equequ1 2048	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧))
4	1, 2, 3	dvelimf-o 39565	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406  ax-c5 39519  ax-c4 39520  ax-c7 39521  ax-c10 39522  ax-c11 39523  ax-c9 39526

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807

This theorem is referenced by:  ax12inda2ALT  39582