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Theorem sbequ1ALT 2580
Description: Alternate version of sbequ1 2250. (Contributed by NM, 16-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
sbequ1ALT (𝑥 = 𝑦 → (𝜑𝜃))

Proof of Theorem sbequ1ALT
StepHypRef Expression
1 pm3.4 809 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 19.8a 2181 . . 3 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 dfsb1.ph . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 586 . 2 ((𝑥 = 𝑦𝜑) → 𝜃)
54ex 416 1 (𝑥 = 𝑦 → (𝜑𝜃))
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  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  sbequ12ALT  2582  dfsb2ALT  2592  sbequiALT  2597  sbi1ALT  2607
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