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Theorem bj-sbievv 34165
Description: Version of sbie 2538 with a second nonfreeness hypothesis and shorter proof. (Contributed by BJ, 18-Jul-2023.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-sbievv.nfx 𝑥𝜓
bj-sbievv.nfy 𝑦𝜑
bj-sbievv.is (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-sbievv ([𝑦 / 𝑥]𝜑𝜓)

Proof of Theorem bj-sbievv
StepHypRef Expression
1 bj-sbievv.nfy . . 3 𝑦𝜑
21sb6f 2531 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
3 bj-sbievv.nfx . . 3 𝑥𝜓
4 bj-sbievv.is . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4equsal 2432 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
62, 5bitri 277 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1528  wnf 1777  [wsb 2062
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 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2383
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by: (None)
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