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Theorem ax12vALT 2505
 Description: Alternate proof of ax12v2 2165, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12vALT (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12vALT
StepHypRef Expression
1 ax-1 6 . . . 4 (𝜑 → (𝑥 = 𝑦𝜑))
2 axc16 2234 . . . 4 (∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
31, 2syl5 34 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
43a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
5 axc15 2387 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
64, 5pm2.61i 177 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828 This theorem is referenced by: (None)
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