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Theorem sb6f 1757
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb6f ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb6f
StepHypRef Expression
1 equs45f.1 . . . 4 (𝜑 → ∀𝑦𝜑)
21sbimi 1720 . . 3 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 sb4a 1755 . . 3 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
42, 3syl 14 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
5 sb2 1723 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
64, 5impbii 125 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-11 1467  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-sb 1719
This theorem is referenced by:  sb5f  1758  sbcof2  1764
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