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Theorem lhpexle1lem 37573
Description: Lemma for lhpexle1 37574 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 485 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
3 simprl 771 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
4 simprr 773 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝜓)
5 simplr 769 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝐴)
6 simpllr 776 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋𝐴)
7 nelne2 3049 . . . . . . 7 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
85, 6, 7syl2anc 588 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
93, 4, 83jca 1126 . . . . 5 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
109ex 417 . . . 4 (((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
1110reximdva 3199 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → (∃𝑝𝐴 (𝑝 𝑊𝜓) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋)))
122, 11mpd 15 . 2 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
131adantr 485 . . 3 ((𝜑 ∧ ¬ 𝑋 𝑊) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
14 simprl 771 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
15 simprr 773 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝜓)
16 simplr 769 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋 𝑊)
17 nbrne2 5050 . . . . . . 7 ((𝑝 𝑊 ∧ ¬ 𝑋 𝑊) → 𝑝𝑋)
1814, 16, 17syl2anc 588 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
1914, 15, 183jca 1126 . . . . 5 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
2019ex 417 . . . 4 ((𝜑 ∧ ¬ 𝑋 𝑊) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
2120reximdv 3198 . . 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 815 1 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 400  w3a 1085  wcel 2112  wne 2952  wrex 3072   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-ral 3076  df-rex 3077  df-v 3412  df-un 3864  df-sn 4521  df-pr 4523  df-op 4527  df-br 5031
This theorem is referenced by:  lhpexle1  37574  lhpexle2  37576  lhpexle3  37578
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