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Theorem sbbid 1726
 Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
sbbid.1 (𝜑 → ∀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 (𝜑 → ∀𝑥𝜑)
2 sbbid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimih 1358 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbbi 1725 . 2 (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
53, 4syl 14 1 (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  sbcomxyyz  1846
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