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Theorem bj-sbimeh 11330
Description: A strengthening of sbieh 1720 (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 1293 . . . 4
21hbth 1397 . . 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 11329 . 2 (⊤ → ([𝑦 / 𝑥]𝜑𝜓))
87mptru 1298 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wtru 1290  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-sb 1693
This theorem is referenced by:  bj-sbime  11331
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