Theorem ax13dgen3 2135

Description: Degenerate instance of ax-13 2372 where bundled variables 𝑦 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)

Assertion

Ref	Expression
ax13dgen3	⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦))

Proof of Theorem ax13dgen3

Step	Hyp	Ref	Expression
1		equid 2015	. . 3 ⊢ 𝑦 = 𝑦
2	1	ax-gen 1798	. 2 ⊢ ∀𝑥 𝑦 = 𝑦
3	2	2a1i 12	1 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)