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Theorem sbc19.21g 3110
 Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
Hypothesis
Ref Expression
sbcgf.1 xφ
Assertion
Ref Expression
sbc19.21g (A V → ([̣A / x]̣(φψ) ↔ (φ → [̣A / xψ)))

Proof of Theorem sbc19.21g
StepHypRef Expression
1 sbcimg 3087 . 2 (A V → ([̣A / x]̣(φψ) ↔ ([̣A / xφ → [̣A / xψ)))
2 sbcgf.1 . . . 4 xφ
32sbcgf 3109 . . 3 (A V → ([̣A / xφφ))
43imbi1d 308 . 2 (A V → (([̣A / xφ → [̣A / xψ) ↔ (φ → [̣A / xψ)))
51, 4bitrd 244 1 (A V → ([̣A / x]̣(φψ) ↔ (φ → [̣A / xψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  [̣wsbc 3046 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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