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Theorem sbrimALT 2611
Description: Alternate version of sbrim 2315. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.s3 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
dfsb1.i2 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
sbrimALT.1 𝑥𝜑
Assertion
Ref Expression
sbrimALT (𝜂 ↔ (𝜑𝜏))

Proof of Theorem sbrimALT
StepHypRef Expression
1 biid 264 . . 3 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 dfsb1.s3 . . 3 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
3 dfsb1.i2 . . 3 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
41, 2, 3sbimALT 2610 . 2 (𝜂 ↔ (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → 𝜏))
5 sbrimALT.1 . . . 4 𝑥𝜑
61, 5sbfALT 2596 . . 3 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ 𝜑)
76imbi1i 353 . 2 ((((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → 𝜏) ↔ (𝜑𝜏))
84, 7bitri 278 1 (𝜂 ↔ (𝜑𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wnf 1785
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-10 2146  ax-12 2179  ax-13 2392
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:  sbiedALT  2616
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