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Theorem sbequ1 1651
 Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sbequ1
StepHypRef Expression
1 pm3.4 316 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 1482 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1646 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 394 . 2 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
54ex 108 1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381  [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-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  sbequ12  1654  sbequi  1720  sb6rf  1733  mo2n  1928  bj-bdfindes  10074  bj-findes  10106
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