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Theorem dfsb2ALT 2591
Description: Alternate version of dfsb2 2532. (Contributed by NM, 17-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.ph (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
dfsb2ALT (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem dfsb2ALT
StepHypRef Expression
1 sp 2183 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 dfsb1.ph . . . . . 6 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
32sbequ2ALT 2580 . . . . 5 (𝑥 = 𝑦 → (𝜃𝜑))
43sps 2185 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜃𝜑))
5 orc 864 . . . 4 ((𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
61, 4, 5syl6an 683 . . 3 (∀𝑥 𝑥 = 𝑦 → (𝜃 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
72sb4ALT 2588 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑)))
8 olc 865 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
97, 8syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
106, 9pm2.61i 185 . 2 (𝜃 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
112sbequ1ALT 2579 . . . 4 (𝑥 = 𝑦 → (𝜑𝜃))
1211imp 410 . . 3 ((𝑥 = 𝑦𝜑) → 𝜃)
132sb2ALT 2587 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜃)
1412, 13jaoi 854 . 2 (((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)) → 𝜃)
1510, 14impbii 212 1 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  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-10 2145  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
This theorem is referenced by:  dfsb3ALT  2592
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