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Theorem bj-sbimeh 14494
Description: A strengthening of sbieh 1790 (same proof). (Contributed by BJ, 16-Dec-2019.)
Hypotheses
Ref Expression
bj-sbimeh.1 (𝜓 → ∀𝑥𝜓)
bj-sbimeh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-sbimeh ([𝑦 / 𝑥]𝜑𝜓)

Proof of Theorem bj-sbimeh
StepHypRef Expression
1 tru 1357 . . . 4
21hbth 1463 . . 3 (⊤ → ∀𝑥⊤)
3 bj-sbimeh.1 . . . 4 (𝜓 → ∀𝑥𝜓)
43a1i 9 . . 3 (⊤ → (𝜓 → ∀𝑥𝜓))
5 bj-sbimeh.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
65a1i 9 . . 3 (⊤ → (𝑥 = 𝑦 → (𝜑𝜓)))
72, 4, 6bj-sbimedh 14493 . 2 (⊤ → ([𝑦 / 𝑥]𝜑𝜓))
87mptru 1362 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351  wtru 1354  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-sb 1763
This theorem is referenced by:  bj-sbime  14495
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