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Theorem bj-spimvwt 32781
Description: Closed form of spimvw 1973. See also spimt 2289. (Contributed by BJ, 8-Nov-2021.)
Assertion
Ref Expression
bj-spimvwt (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-spimvwt
StepHypRef Expression
1 bj-alequexv 32780 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ∃𝑥(𝜑𝜓))
2 19.36v 1960 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
31, 2sylib 208 1 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945
This theorem depends on definitions:  df-bi 197  df-ex 1745
This theorem is referenced by: (None)
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