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Theorem sbanALT 2610
 Description: Alternate version of sban 2086. (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.an (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
Assertion
Ref Expression
sbanALT (𝜂 ↔ (𝜃𝜏))

Proof of Theorem sbanALT
StepHypRef Expression
1 biid 263 . . . 4 (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ ((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))))
2 biid 263 . . . 4 (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))))
31, 2sbnALT 2595 . . 3 (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ¬ ((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))))
4 dfsb1.p6 . . . . 5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
5 biid 263 . . . . 5 (((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)) ↔ ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)))
64, 5, 1sbimALT 2608 . . . 4 (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ (𝜃 → ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓))))
7 dfsb1.s4 . . . . . 6 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
87, 5sbnALT 2595 . . . . 5 (((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓)) ↔ ¬ 𝜏)
98imbi2i 338 . . . 4 ((𝜃 → ((𝑥 = 𝑦 → ¬ 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜓))) ↔ (𝜃 → ¬ 𝜏))
106, 9bitri 277 . . 3 (((𝑥 = 𝑦 → (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 → ¬ 𝜓))) ↔ (𝜃 → ¬ 𝜏))
113, 10xchbinx 336 . 2 (((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))) ↔ ¬ (𝜃 → ¬ 𝜏))
12 dfsb1.an . . 3 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
13 df-an 399 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
1412, 2, 13sbbiiALT 2578 . 2 (𝜂 ↔ ((𝑥 = 𝑦 → ¬ (𝜑 → ¬ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ (𝜑 → ¬ 𝜓))))
15 df-an 399 . 2 ((𝜃𝜏) ↔ ¬ (𝜃 → ¬ 𝜏))
1611, 14, 153bitr4i 305 1 (𝜂 ↔ (𝜃𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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:  sbbiALT  2611
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