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Theorem sbi1v 2337
 Description: Move implication out of substitution. Version of sbi1 2523 with a disjoint variable condition, not requiring ax-13 2391. (Contributed by Wolf Lammen, 18-Jan-2023.)
Assertion
Ref Expression
sbi1v ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbi1v
StepHypRef Expression
1 sb4v 2308 . 2 ([𝑦 / 𝑥](𝜑𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 sb4v 2308 . . 3 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
3 ax-2 7 . . . 4 ((𝑥 = 𝑦 → (𝜑𝜓)) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
43al2imi 1916 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
5 sb2v 2302 . . 3 (∀𝑥(𝑥 = 𝑦𝜓) → [𝑦 / 𝑥]𝜓)
62, 4, 5syl56 36 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
71, 6syl 17 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1656  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-12 2222 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070 This theorem is referenced by:  sbimv  2339  spsbimvOLD  2343
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