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Theorem sbi1ALT 2606
Description: Alternate version of sbi1 2076. (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
sbi1ALT (𝜂 → (𝜃𝜏))

Proof of Theorem sbi1ALT
StepHypRef Expression
1 dfsb1.p5 . . . . . 6 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21sbequ2ALT 2580 . . . . 5 (𝑥 = 𝑦 → (𝜃𝜑))
3 dfsb1.im . . . . . 6 (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑𝜓))))
43sbequ2ALT 2580 . . . . 5 (𝑥 = 𝑦 → (𝜂 → (𝜑𝜓)))
52, 4syl5d 73 . . . 4 (𝑥 = 𝑦 → (𝜂 → (𝜃𝜓)))
6 dfsb1.s2 . . . . 5 (𝜏 ↔ ((𝑥 = 𝑦𝜓) ∧ ∃𝑥(𝑥 = 𝑦𝜓)))
76sbequ1ALT 2579 . . . 4 (𝑥 = 𝑦 → (𝜓𝜏))
85, 7syl6d 75 . . 3 (𝑥 = 𝑦 → (𝜂 → (𝜃𝜏)))
98sps 2185 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜂 → (𝜃𝜏)))
101sb4ALT 2588 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥(𝑥 = 𝑦𝜑)))
113sb4ALT 2588 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜂 → ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓))))
12 ax-2 7 . . . . . 6 ((𝑥 = 𝑦 → (𝜑𝜓)) → ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜓)))
1312al2imi 1817 . . . . 5 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜓)))
146sb2ALT 2587 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜓) → 𝜏)
1513, 14syl6 35 . . . 4 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜏))
1611, 15syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜂 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜏)))
1710, 16syl5d 73 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜂 → (𝜃𝜏)))
189, 17pm2.61i 185 1 (𝜂 → (𝜃𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  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 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
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  2608
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