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Theorem sbequ2ALT 2580
Description: Alternate version of sbequ2 2251. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
sbequ2ALT (𝑥 = 𝑦 → (𝜃𝜑))

Proof of Theorem sbequ2ALT
StepHypRef Expression
1 dfsb1.ph . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simplbi 501 . 2 (𝜃 → (𝑥 = 𝑦𝜑))
32com12 32 1 (𝑥 = 𝑦 → (𝜃𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by:  sbequ12ALT  2581  dfsb2ALT  2591  sbequiALT  2596  sbi1ALT  2606
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