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Theorem sbi2ALT 2609
Description: Alternate version of sbi2 2312. (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.p5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
dfsb1.s2 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
dfsb1.im (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
Assertion
Ref Expression
sbi2ALT ((𝜃𝜏) → 𝜂)

Proof of Theorem sbi2ALT
StepHypRef Expression
1 dfsb1.p5 . . . 4 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 biid 264 . . . 4 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
31, 2sbnALT 2597 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ¬ 𝜃)
4 dfsb1.im . . . 4 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
5 pm2.21 123 . . . 4 𝜑 → (𝜑𝜓))
62, 4, 5sbimiALT 2579 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) → 𝜂)
73, 6sylbir 238 . 2 𝜃𝜂)
8 dfsb1.s2 . . 3 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
9 ax-1 6 . . 3 (𝜓 → (𝜑𝜓))
108, 4, 9sbimiALT 2579 . 2 (𝜏𝜂)
117, 10ja 189 1 ((𝜃𝜏) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wex 1781
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:  sbimALT  2610
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