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Theorem sbbiALT 2611
 Description: Alternate version of sbbi 2317. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.p6 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
dfsb1.s4 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
dfsb1.bi (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
Assertion
Ref Expression
sbbiALT (𝜂 ↔ (𝜃𝜏))

Proof of Theorem sbbiALT
StepHypRef Expression
1 dfsb1.bi . . 3 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
2 biid 263 . . 3 (((𝑥 = 𝑦 → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑𝜓) ∧ (𝜓𝜑)))) ↔ ((𝑥 = 𝑦 → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑𝜓) ∧ (𝜓𝜑)))))
3 dfbi2 477 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
41, 2, 3sbbiiALT 2578 . 2 (𝜂 ↔ ((𝑥 = 𝑦 → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑𝜓) ∧ (𝜓𝜑)))))
5 dfsb1.p6 . . . . 5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
6 dfsb1.s4 . . . . 5 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
7 biid 263 . . . . 5 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
85, 6, 7sbimALT 2608 . . . 4 (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ↔ (𝜃𝜏))
9 biid 263 . . . . 5 (((𝑥 = 𝑦 → (𝜓𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓𝜑))) ↔ ((𝑥 = 𝑦 → (𝜓𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓𝜑))))
106, 5, 9sbimALT 2608 . . . 4 (((𝑥 = 𝑦 → (𝜓𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓𝜑))) ↔ (𝜏𝜃))
118, 10anbi12i 628 . . 3 ((((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ∧ ((𝑥 = 𝑦 → (𝜓𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓𝜑)))) ↔ ((𝜃𝜏) ∧ (𝜏𝜃)))
127, 9, 2sbanALT 2610 . . 3 (((𝑥 = 𝑦 → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑𝜓) ∧ (𝜓𝜑)))) ↔ (((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))) ∧ ((𝑥 = 𝑦 → (𝜓𝜑)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜓𝜑)))))
13 dfbi2 477 . . 3 ((𝜃𝜏) ↔ ((𝜃𝜏) ∧ (𝜏𝜃)))
1411, 12, 133bitr4i 305 . 2 (((𝑥 = 𝑦 → ((𝜑𝜓) ∧ (𝜓𝜑))) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ((𝜑𝜓) ∧ (𝜓𝜑)))) ↔ (𝜃𝜏))
154, 14bitri 277 1 (𝜂 ↔ (𝜃𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1780 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:  sblbisALT  2612
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