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Theorem bj-equsal 32001
 Description: Shorter proof of equsal 2279. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2279, 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 218 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3bj-equsal1 31999 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜓)
52biimprd 237 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
61, 5bj-equsal2 32000 . 2 (𝜓 → ∀𝑥(𝑥 = 𝑦𝜑))
74, 6impbii 198 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
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