Theorem ax11i 1692

Description: Inference that has ax-11 1484 (without ∀𝑦) as its conclusion and doesn't require ax-10 1483, ax-11 1484, or ax-12 1489 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

Hypotheses

Ref	Expression
ax11i.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
ax11i.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
ax11i	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem ax11i

Step	Hyp	Ref	Expression
1		ax11i.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		ax11i.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3	1	biimprcd 159	. . 3 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑))
4	2, 3	alrimih 1445	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 4	syl6bi 162	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)