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Theorem ax12wlem 1996
 Description: Lemma for weak version of ax-12 2034. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1997. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax12wlemw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
ax12wlem (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem ax12wlem
StepHypRef Expression
1 ax12wlemw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
2 ax-5 1827 . 2 (𝜓 → ∀𝑥𝜓)
31, 2ax12i 1866 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  ax-5 1827 This theorem depends on definitions:  df-bi 196 This theorem is referenced by:  ax12w  1997
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