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Theorem sbrim 2239
Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sbrim.1 𝑥𝜑
Assertion
Ref Expression
sbrim ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbrim
StepHypRef Expression
1 sbim 2238 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbrim.1 . . . 4 𝑥𝜑
32sbf 2200 . . 3 ([𝑦 / 𝑥]𝜑𝜑)
43imbi1i 342 . 2 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
51, 4bitri 267 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wnf 1747  [wsb 2016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-sb 2017
This theorem is referenced by:  sbied  2470  sbco2d  2479  2mos  2680
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