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Theorem sbiedALT 2608
Description: Alternate version of sbied 2539. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.s5 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
sbiedALT.1 𝑥𝜑
sbiedALT.2 (𝜑 → Ⅎ𝑥𝜒)
sbiedALT.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
sbiedALT (𝜑 → (𝜏𝜒))

Proof of Theorem sbiedALT
StepHypRef Expression
1 dfsb1.s5 . . . 4 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
2 biid 263 . . . 4 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
3 sbiedALT.1 . . . 4 𝑥𝜑
41, 2, 3sbrimALT 2603 . . 3 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ (𝜑𝜏))
5 sbiedALT.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
63, 5nfim1 2191 . . . 4 𝑥(𝜑𝜒)
7 sbiedALT.3 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98pm5.74d 275 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ (𝜑𝜒)))
102, 6, 9sbieALT 2607 . . 3 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ (𝜑𝜒))
114, 10bitr3i 279 . 2 ((𝜑𝜏) ↔ (𝜑𝜒))
1211pm5.74ri 274 1 (𝜑 → (𝜏𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2383
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by:  sbco2ALT  2609
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