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Theorem sbrimv 2311
 Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2310 not depending on ax-10 2143, but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024.)
Hypothesis
Ref Expression
sbrim.1 𝑥𝜑
Assertion
Ref Expression
sbrimv ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbrimv
StepHypRef Expression
1 sbrim.1 . . 3 𝑥𝜑
2119.21 2206 . 2 (∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))
32sbrimvlem 2099 1 ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1786  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by:  sbiedw  2324
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