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Theorem sbbid 2391
 Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
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 2069 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbbi 2390 . 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 195  ∀wal 1473  Ⅎwnf 1699  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  sbcom3  2399  sbco3  2405  sbcom2  2433  sbal  2450  wl-equsb3  32516  wl-sbcom2d-lem1  32521  wl-sbcom3  32551
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