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Theorem sbequ1 2277
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.)
Assertion
Ref Expression
sbequ1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sbequ1
StepHypRef Expression
1 pm3.4 845 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 2216 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 2065 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 579 . 2 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
54ex 402 1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wex 1875  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-sb 2065
This theorem is referenced by:  sbequ12  2278  dfsb2  2490  sbequi  2492  sbi1  2509  2eu6  2714  sb5ALT  39511  2pm13.193  39538  2pm13.193VD  39899  sb5ALTVD  39909
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