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Theorem bj-ax12v3ALT 34028
Description: Alternate proof of bj-ax12v3 34027. Uses axc11r 2386 and axc15 2444 instead of ax-12 2177. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ax12v3ALT (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-ax12v3ALT
StepHypRef Expression
1 ax-5 1911 . . . 4 (𝜑 → ∀𝑦𝜑)
2 axc11r 2386 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
3 ala1 1814 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
41, 2, 3syl56 36 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
54a1d 25 . 2 (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
6 axc15 2444 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
75, 6pm2.61i 184 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785
This theorem is referenced by: (None)
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