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Theorem bj-equsal1t 33138
Description: Duplication of wl-equsal1t 33655, with shorter proof. If one imposes a DV condition on x,y , then one can use bj-alequexv 32986 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33656 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 33039 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
2 19.9t 2226 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
31, 2syl5ib 234 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
4 nf5r 2217 . . 3 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5 ala1 1888 . . 3 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl6 35 . 2 (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
73, 6impbid 202 1 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wex 1851  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202  ax-13 2407
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by:  bj-equsal1ti  33139
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