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Theorem sb6f 2039
 Description: Equivalence for substitution when y is not free in φ. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sb6f.1 yφ
Assertion
Ref Expression
sb6f ([y / x]φx(x = yφ))

Proof of Theorem sb6f
StepHypRef Expression
1 sb6f.1 . . . . 5 yφ
21nfri 1762 . . . 4 (φyφ)
32sbimi 1652 . . 3 ([y / x]φ → [y / x]yφ)
4 sb4a 1923 . . 3 ([y / x]yφx(x = yφ))
53, 4syl 15 . 2 ([y / x]φx(x = yφ))
6 sb2 2023 . 2 (x(x = yφ) → [y / x]φ)
75, 6impbii 180 1 ([y / x]φx(x = yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  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:  sb5f  2040
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