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Theorem sbrim 1827
 Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
sbrim.1 (φxφ)
Assertion
Ref Expression
sbrim ([y / x](φψ) ↔ (φ → [y / x]ψ))

Proof of Theorem sbrim
StepHypRef Expression
1 sbim 1824 . 2 ([y / x](φψ) ↔ ([y / x]φ → [y / x]ψ))
2 sbrim.1 . . . 4 (φxφ)
32sbh 1656 . . 3 ([y / x]φφ)
43imbi1i 227 . 2 (([y / x]φ → [y / x]ψ) ↔ (φ → [y / x]ψ))
51, 4bitri 173 1 ([y / x](φψ) ↔ (φ → [y / x]ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by:  sbco2d  1837  sbco2vd  1838  hbsbd  1855
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