Theorem ax12f 38882

Description: Basis step for constructing a substitution instance of ax-c15 38831 without using ax-c15 38831. We can start with any formula 𝜑 in which 𝑥 is not free. (Contributed by NM, 21-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax12f.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
ax12f	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

Proof of Theorem ax12f

Step	Hyp	Ref	Expression
1		ax12f.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-1 6	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
3	1, 2	alrimih 1823	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	3	2a1i 12	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1794  ax-4 1808

This theorem is referenced by: (None)