Theorem ax12dgen 2172

Description: Degenerate instance of ax-12 2216 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 1846	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑))
2	1	a1i 11	1 ⊢ (𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-gen 1828  ax-4 1842

This theorem is used by: (None)