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Theorem lhpexle1lem 36028
Description: Lemma for lhpexle1 36029 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 473 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
3 simprl 788 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
4 simprr 790 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝜓)
5 simplr 786 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝐴)
6 simpllr 794 . . . . . . 7 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋𝐴)
7 nelne2 3068 . . . . . . 7 ((𝑝𝐴 ∧ ¬ 𝑋𝐴) → 𝑝𝑋)
85, 6, 7syl2anc 580 . . . . . 6 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
93, 4, 83jca 1159 . . . . 5 ((((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
109ex 402 . . . 4 (((𝜑 ∧ ¬ 𝑋𝐴) ∧ 𝑝𝐴) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
1110reximdva 3197 . . 3 ((𝜑 ∧ ¬ 𝑋𝐴) → (∃𝑝𝐴 (𝑝 𝑊𝜓) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋)))
122, 11mpd 15 . 2 ((𝜑 ∧ ¬ 𝑋𝐴) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
131adantr 473 . . 3 ((𝜑 ∧ ¬ 𝑋 𝑊) → ∃𝑝𝐴 (𝑝 𝑊𝜓))
14 simprl 788 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝 𝑊)
15 simprr 790 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝜓)
16 simplr 786 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → ¬ 𝑋 𝑊)
17 nbrne2 4863 . . . . . . 7 ((𝑝 𝑊 ∧ ¬ 𝑋 𝑊) → 𝑝𝑋)
1814, 16, 17syl2anc 580 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → 𝑝𝑋)
1914, 15, 183jca 1159 . . . . 5 (((𝜑 ∧ ¬ 𝑋 𝑊) ∧ (𝑝 𝑊𝜓)) → (𝑝 𝑊𝜓𝑝𝑋))
2019ex 402 . . . 4 ((𝜑 ∧ ¬ 𝑋 𝑊) → ((𝑝 𝑊𝜓) → (𝑝 𝑊𝜓𝑝𝑋)))
2120reximdv 3196 . . 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 850 1 (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  w3a 1108  wcel 2157  wne 2971  wrex 3090   class class class wbr 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844
This theorem is referenced by:  lhpexle1  36029  lhpexle2  36031  lhpexle3  36033
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