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Theorem sbco2ALT 2615
 Description: Alternate version of sbco2 2553. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.p8 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
dfsb1.sb (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))))
sbco2ALT.1 𝑧𝜑
Assertion
Ref Expression
sbco2ALT (𝜏𝜃)

Proof of Theorem sbco2ALT
StepHypRef Expression
1 dfsb1.sb . . . . 5 (𝜏 ↔ ((𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑))) ∧ ∃𝑧(𝑧 = 𝑦 ∧ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))))
21sbequ12ALT 2581 . . . 4 (𝑧 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ 𝜏))
3 biid 263 . . . . 5 (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ ((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)))
4 dfsb1.p8 . . . . 5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
53, 4sbequALT 2597 . . . 4 (𝑧 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ 𝜃))
62, 5bitr3d 283 . . 3 (𝑧 = 𝑦 → (𝜏𝜃))
76sps 2184 . 2 (∀𝑧 𝑧 = 𝑦 → (𝜏𝜃))
8 nfnae 2456 . . 3 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9 sbco2ALT.1 . . . 4 𝑧𝜑
104, 9nfsb4ALT 2605 . . 3 (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃)
115a1i 11 . . 3 (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → (((𝑥 = 𝑧𝜑) ∧ ∃𝑥(𝑥 = 𝑧𝜑)) ↔ 𝜃)))
121, 8, 10, 11sbiedALT 2614 . 2 (¬ ∀𝑧 𝑧 = 𝑦 → (𝜏𝜃))
137, 12pm2.61i 184 1 (𝜏𝜃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃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-11 2161  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785 This theorem is referenced by:  sb7fALT  2616
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