Theorem ax12dgen1 1725

Description: Degenerate instance of ax-12 1925 where bundled variables x and y have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)

Assertion

Ref	Expression
ax12dgen1	⊢ (¬ x = x → (x = z → ∀x x = z))

Proof of Theorem ax12dgen1

Step	Hyp	Ref	Expression
1		equid 1676	. 2 ⊢ x = x
2	1	pm2.24i 136	1 ⊢ (¬ x = x → (x = z → ∀x x = z))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ax12dgen4  1728