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Theorem ceqsal1t 3499
Description: One direction of ceqsalt 3500 is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft 3535. Extracted from a proof of ceqsalt 3500. (Contributed by Wolf Lammen, 25-Mar-2025.)
Assertion
Ref Expression
ceqsal1t ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))

Proof of Theorem ceqsal1t
StepHypRef Expression
1 biimpr 219 . . . . . 6 ((𝜑𝜓) → (𝜓𝜑))
21imim2i 16 . . . . 5 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
32com23 86 . . . 4 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
43alimi 1805 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
5 19.21t 2191 . . 3 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
64, 5imbitrid 243 . 2 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
76imp 406 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778
This theorem is referenced by:  ceqsalt  3500
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