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Theorem sbimd 2243
 Description: Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2070. (Revised by Steven Nguyen, 9-Jul-2023.)
Hypotheses
Ref Expression
sbimd.1 𝑥𝜑
sbimd.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbimd (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))

Proof of Theorem sbimd
StepHypRef Expression
1 sbimd.1 . . 3 𝑥𝜑
2 sbimd.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2211 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbim 2077 . 2 (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
53, 4syl 17 1 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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