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Theorem lhpexle1lem 34311
Description: Lemma for lhpexle1 34312 and others that eliminates restrictions on 𝑋. (Contributed by NM, 24-Jul-2013.)
Hypotheses
Ref Expression
lhpexle1lem.1 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓))
lhpexle1lem.2 ((𝜑 ∧ (𝑋𝐴𝑋 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Assertion
Ref Expression
lhpexle1lem (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Distinct variable groups:   ,𝑝   𝐴,𝑝   𝑊,𝑝   𝑋,𝑝   𝜑,𝑝
Allowed substitution hint:   𝜓(𝑝)

Proof of Theorem lhpexle1lem
StepHypRef Expression
1 lhpexle1lem.1 . . . 4 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓))
21adantr 480 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
3 simprl 790 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
4 simprr 792 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝜓)
5 simplr 788 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝐴)
6 simpllr 795 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋𝐴)
7 nelne2 2879 . . . . . . 7 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
85, 6, 7syl2anc 691 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
93, 4, 83jca 1235 . . . . 5 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
109ex 449 . . . 4 (((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
1110reximdva 3000 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → (∃𝑝𝐴 (𝑝 𝑊𝜓) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋)))
122, 11mpd 15 . 2 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
131adantr 480 . . 3 ((𝜑 ∧ ¬ 𝑋 𝑊) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
14 simprl 790 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
15 simprr 792 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝜓)
16 simplr 788 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋 𝑊)
17 nbrne2 4603 . . . . . . 7 ((𝑝 𝑊 ∧ ¬ 𝑋 𝑊) → 𝑝𝑋)
1814, 16, 17syl2anc 691 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
1914, 15, 183jca 1235 . . . . 5 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
2019ex 449 . . . 4 ((𝜑 ∧ ¬ 𝑋 𝑊) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
2120reximdv 2999 . . 3 ((𝜑 ∧ ¬ 𝑋 𝑊) → (∃𝑝𝐴 (𝑝 𝑊𝜓) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋)))
2213, 21mpd 15 . 2 ((𝜑 ∧ ¬ 𝑋 𝑊) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
23 lhpexle1lem.2 . 2 ((𝜑 ∧ (𝑋𝐴𝑋 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
2412, 22, 23pm2.61dda 830 1 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031  wcel 1977  wne 2780  wrex 2897   class class class wbr 4583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584
This theorem is referenced by:  lhpexle1  34312  lhpexle2  34314  lhpexle3  34316
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