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Theorem sbbid 2242
Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2141 and ax-13 2386. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2066. (Revised by Steven Nguyen, 11-Jul-2023.)
Hypotheses
Ref Expression
sbbid.1 𝑥𝜑
sbbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbbid (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Proof of Theorem sbbid
StepHypRef Expression
1 sbbid.1 . . 3 𝑥𝜑
2 sbbid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2209 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbbi 2074 . 2 (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
53, 4syl 17 1 (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531  wnf 1780  [wsb 2065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781  df-sb 2066
This theorem is referenced by:  2sbbid  2243  sbcom3  2544  sbco3  2551  sbalOLD  2571  wl-sbcom2d-lem1  34794  wl-2spsbbi  34800  wl-clabt  34829  dfich2bi  43614
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