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Theorem sbequ1OLD 2521
Description: Obsolete version of sbequ1 2250 as of 8-Jul-2023. An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbequ1OLD (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Proof of Theorem sbequ1OLD
StepHypRef Expression
1 pm3.4 809 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 2181 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 dfsb1 2511 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 586 . 2 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
54ex 416 1 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1781  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071
This theorem is referenced by: (None)
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