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Theorem bj-axc10 31729
Description: Alternate (shorter) proof of axc10 2143. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc10 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Proof of Theorem bj-axc10
StepHypRef Expression
1 ax6e 2141 . . 3 𝑥 𝑥 = 𝑦
2 exim 1739 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝑥𝜑))
31, 2mpi 20 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥𝑥𝜑)
4 axc7e 1993 . 2 (∃𝑥𝑥𝜑𝜑)
53, 4syl 17 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by: (None)
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