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Theorem bj-equsal1t 34261
Description: Duplication of wl-equsal1t 34945, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2007 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 34946 is also interesting. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-equsal1t (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))

Proof of Theorem bj-equsal1t
StepHypRef Expression
1 bj-alequex 34222 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
2 19.9t 2203 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2syl5ib 247 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
4 nf5r 2192 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 ala1 1815 . . 3 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl6 35 . 2 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
73, 6impbid 215 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  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 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  bj-equsal1ti  34262
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