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Theorem ax12i 1866
 Description: Inference that has ax-12 2034 (without ∀𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2034 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 → (𝜑𝜓))
ax12i.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
ax12i (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12i
StepHypRef Expression
1 ax12i.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
2 ax12i.2 . . 3 (𝜓 → ∀𝑥𝜓)
31biimprcd 239 . . 3 (𝜓 → (𝑥 = 𝑦𝜑))
42, 3alrimih 1741 . 2 (𝜓 → ∀𝑥(𝑥 = 𝑦𝜑))
51, 4syl6bi 242 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196 This theorem is referenced by:  ax12wlem  1996
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