Theorem ax13dgen4 2138

Description: Degenerate instance of ax-13 2384 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Therefore, also a degenerate instance of ax13dgen1 2135, ax13dgen2 2136, and ax13dgen3 2137. Also an instance of the intuitionistic tautology pm2.21 123. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)

Assertion

Ref	Expression
ax13dgen4	⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Proof of Theorem ax13dgen4

Step	Hyp	Ref	Expression
1		pm2.21 123	1 ⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)