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Theorem sbequ1 2255
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 pm3.4 585 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 2197 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 df-sb 2045 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 701 . 2 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
54ex 449 1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1851  [wsb 2044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-12 2194
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1852  df-sb 2045
This theorem is referenced by:  sbequ12  2256  dfsb2  2508  sbequi  2510  sbi1  2527  2eu6  2694  sb5ALT  39231  2pm13.193  39268  2pm13.193VD  39636  sb5ALTVD  39646
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