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Theorem sb6rf 1778
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb6rf (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 sbequ1 1695 . . . . 5 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
32equcoms 1638 . . . 4 (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
43com12 30 . . 3 (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
51, 4alrimih 1401 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6 sb2 1694 . . 3 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
71sbid2h 1774 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
86, 7sylib 120 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑)
95, 8impbii 124 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1285  [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-sb 1690
This theorem is referenced by:  2sb6rf  1911  eu1  1970
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