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Theorem bj-equsal 34264
Description: Shorter proof of equsal 2428. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2428, but "min */exc equsal" is ok. (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-equsal.1 𝑥𝜓
bj-equsal.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-equsal (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem bj-equsal
StepHypRef Expression
1 bj-equsal.1 . . 3 𝑥𝜓
2 bj-equsal.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 232 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3bj-equsal1 34262 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)
52biimprd 251 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
61, 5bj-equsal2 34263 . 2 (𝜓 → ∀𝑥(𝑥 = 𝑦𝜑))
74, 6impbii 212 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175  ax-13 2379
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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