Theorem ax12dgen 2133

Description: Degenerate instance of ax-12 2176 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
ax12dgen	⊢ (𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑)))

Proof of Theorem ax12dgen

Step	Hyp	Ref	Expression
1		ala1 1812	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑))
2	1	a1i 11	1 ⊢ (𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1794  ax-4 1808

This theorem is referenced by: (None)