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Theorem sbieALT 2613
 Description: Alternate version of sbie 2544. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.p7 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
sbieALT.1 𝑥𝜓
sbieALT.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbieALT (𝜃𝜓)

Proof of Theorem sbieALT
StepHypRef Expression
1 biid 263 . . . 4 (((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)) ↔ ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)))
21equsb1ALT 2601 . . 3 ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦))
3 biid 263 . . . 4 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
4 sbieALT.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
51, 3, 4sbimiALT 2577 . . 3 (((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)) → ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
62, 5ax-mp 5 . 2 ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓)))
7 dfsb1.p7 . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
8 biid 263 . . 3 (((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)) ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
9 sbieALT.1 . . . 4 𝑥𝜓
108, 9sbfALT 2594 . . 3 (((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)) ↔ 𝜓)
117, 8, 3, 10sblbisALT 2612 . 2 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ (𝜃𝜓))
126, 11mpbi 232 1 (𝜃𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1780  Ⅎwnf 1784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785 This theorem is referenced by:  sbiedALT  2614
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