Theorem dveeq1-o16 37611

Description: Version of dveeq1 2378 using ax-c16 37567 instead of ax-5 1913. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1913. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dveeq1-o16	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o16

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax5eq 37607	. 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2		ax5eq 37607	. 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3		equequ1 2028	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧))
4	1, 2, 3	dvelimh 2448	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-c9 37565  ax-c16 37567

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786

This theorem is referenced by: (None)