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Theorem ax12i 1876
Description: Inference that has ax-12 2044 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2044 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 240 . . 3 (𝜓 → (𝑥 = 𝑦𝜑))
42, 3alrimih 1748 . 2 (𝜓 → ∀𝑥(𝑥 = 𝑦𝜑))
51, 4syl6bi 243 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  ax12wlem  2006
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