Theorem ax11i 1642

Description: Inference that has ax-11 1437 (without ∀𝑦) as its conclusion and doesn't require ax-10 1436, ax-11 1437, or ax-12 1442 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 158	. . 3 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑))
4	2, 3	alrimih 1398	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 4	syl6bi 161	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)