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Theorem lhpexle1lem 37010
Description: Lemma for lhpexle1 37011 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 481 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
3 simprl 767 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
4 simprr 769 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝜓)
5 simplr 765 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝐴)
6 simpllr 772 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋𝐴)
7 nelne2 3120 . . . . . . 7 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
85, 6, 7syl2anc 584 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
93, 4, 83jca 1122 . . . . 5 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
109ex 413 . . . 4 (((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
1110reximdva 3279 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → (∃𝑝𝐴 (𝑝 𝑊𝜓) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋)))
122, 11mpd 15 . 2 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
131adantr 481 . . 3 ((𝜑 ∧ ¬ 𝑋 𝑊) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
14 simprl 767 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
15 simprr 769 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝜓)
16 simplr 765 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋 𝑊)
17 nbrne2 5083 . . . . . . 7 ((𝑝 𝑊 ∧ ¬ 𝑋 𝑊) → 𝑝𝑋)
1814, 16, 17syl2anc 584 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
1914, 15, 183jca 1122 . . . . 5 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
2019ex 413 . . . 4 ((𝜑 ∧ ¬ 𝑋 𝑊) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
2120reximdv 3278 . . 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 811 1 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064
This theorem is referenced by:  lhpexle1  37011  lhpexle2  37013  lhpexle3  37015
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