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Theorem sb2ALT 2587
Description: Alternate version of sb2 2505. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
sb2ALT (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)

Proof of Theorem sb2ALT
StepHypRef Expression
1 sp 2183 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
2 equs4 2439 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
3 dfsb1.ph . 2 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
41, 2, 3sylanbrc 586 1 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  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 1911  ax-6 1970  ax-7 2015  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  stdpc4ALT  2590  dfsb2ALT  2591  sbequiALT  2596  hbsb2ALT  2599  equsb1ALT  2601  sb6fALT  2602  sbi1ALT  2606
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