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Theorem sbi2 2064
 Description: Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbi2 (([y / x]φ → [y / x]ψ) → [y / x](φψ))

Proof of Theorem sbi2
StepHypRef Expression
1 sbn 2062 . . 3 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
2 pm2.21 100 . . . 4 φ → (φψ))
32sbimi 1652 . . 3 ([y / x] ¬ φ → [y / x](φψ))
41, 3sylbir 204 . 2 (¬ [y / x]φ → [y / x](φψ))
5 ax-1 6 . . 3 (ψ → (φψ))
65sbimi 1652 . 2 ([y / x]ψ → [y / x](φψ))
74, 6ja 153 1 (([y / x]φ → [y / x]ψ) → [y / x](φψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  [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:  sbim  2065
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