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Theorem bj-sbimeh 13357
Description: A strengthening of sbieh 1770 (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 1339 . . . 4
21hbth 1443 . . 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 13356 . 2 (⊤ → ([𝑦 / 𝑥]𝜑𝜓))
87mptru 1344 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wtru 1336  [wsb 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-sb 1743
This theorem is referenced by:  bj-sbime  13358
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