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Theorem sbequ1 1722
 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 pm3.4 329 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 1550 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 1717 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 411 . 2 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
54ex 114 1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1449  [wsb 1716 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468 This theorem depends on definitions:  df-bi 116  df-sb 1717 This theorem is referenced by:  sbequ12  1725  sbequi  1791  sb6rf  1805  mo2n  2001  bj-bdfindes  12830  bj-findes  12862
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