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Theorem wl-sblimt 33462
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2424. (Contributed by Wolf Lammen, 26-Jul-2019.)
Assertion
Ref Expression
wl-sblimt (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))

Proof of Theorem wl-sblimt
StepHypRef Expression
1 sbim 2423 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2 sbft 2407 . . 3 (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥]𝜓𝜓))
32imbi2d 329 . 2 (Ⅎ𝑥𝜓 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
41, 3syl5bb 272 1 (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1748  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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