Theorem ax12i 1963

Description: Inference that has ax-12 2170 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2170 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)

Hypotheses

Ref	Expression
ax12i.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
ax12i.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
ax12i	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem ax12i

Step	Hyp	Ref	Expression
1		ax12i.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		ax12i.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3	1	biimprcd 252	. . 3 ⊢ (𝜓 → (𝑥 = 𝑦 → 𝜑))
4	2, 3	alrimih 1818	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 4	syl6bi 255	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804

This theorem depends on definitions:  df-bi 209

This theorem is referenced by:  ax12wlem  2130