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Theorem bj-axc10 32832
 Description: Alternate (shorter) proof of axc10 2288. One can prove a version with DV(x,y) without ax-13 2282, by using ax6ev 1947 instead of ax6e 2286. (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 2286 . . 3 𝑥 𝑥 = 𝑦
2 exim 1801 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝑥𝜑))
31, 2mpi 20 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥𝑥𝜑)
4 axc7e 2171 . 2 (∃𝑥𝑥𝜑𝜑)
53, 4syl 17 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by: (None)
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