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Theorem bj-equsal2 32455
Description: One direction of equsal 2290. (Contributed by BJ, 30-Sep-2018.)
Hypotheses
Ref Expression
bj-equsal2.1 𝑥𝜑
bj-equsal2.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-equsal2 (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))

Proof of Theorem bj-equsal2
StepHypRef Expression
1 bj-equsal2.1 . . 3 𝑥𝜑
21bj-equsal1ti 32453 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
3 bj-equsal2.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43a2i 14 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓))
54alimi 1736 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓))
62, 5sylbir 225 1 (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-equsal  32456
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