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Theorem sb6rf 1733
 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 1651 . . . . 5 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
32equcoms 1594 . . . 4 (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
43com12 27 . . 3 (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
51, 4alrimih 1358 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6 sb2 1650 . . 3 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
71sbid2h 1729 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑𝜑)
86, 7sylib 127 . 2 (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑)
95, 8impbii 117 1 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  2sb6rf  1866  eu1  1925
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