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Theorem sbrim 2067
 Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-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 2065 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
2 sbrim.1 . . . 4 xφ
32sbf 2026 . . 3 ([y / x]φφ)
43imbi1i 315 . 2 (([y / x]φ → [y / x]ψ) ↔ (φ → [y / x]ψ))
51, 4bitri 240 1 ([y / x](φψ) ↔ (φ → [y / x]ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbco2d  2087  2mos  2283
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