Theorem dveeq1-o16 36687

Description: Version of dveeq1 2379 using ax-c16 36643 instead of ax-5 1918. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1918. (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 36683	. 2 ⊢ (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2		ax5eq 36683	. 2 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3		equequ1 2033	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑦 = 𝑧))
4	1, 2, 3	dvelimh 2449	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-c9 36641  ax-c16 36643

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)