Theorem lhpexle1lem 40009

Description: Lemma for lhpexle1 40010 and others that eliminates restrictions on 𝑋. (Contributed by NM, 24-Jul-2013.)

Hypotheses

Ref	Expression
lhpexle1lem.1	⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓))
lhpexle1lem.2	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))

Assertion

Ref	Expression
lhpexle1lem	⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))

Distinct variable groups:   ≤ ,𝑝   𝐴,𝑝   𝑊,𝑝   𝑋,𝑝   𝜑,𝑝

Allowed substitution hint:   𝜓(𝑝)

Proof of Theorem lhpexle1lem

Step	Hyp	Ref	Expression
1		lhpexle1lem.1	. . . 4 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓))
3		simprl 771	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝑝 ≤ 𝑊)
4		simprr 773	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝜓)
5		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝑝 ∈ 𝐴)
6		simpllr 776	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → ¬ 𝑋 ∈ 𝐴)
7		nelne2 3040	. . . . . . 7 ⊢ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝐴) → 𝑝 ≠ 𝑋)
8	5, 6, 7	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝑝 ≠ 𝑋)
9	3, 4, 8	3jca 1129	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
10	9	ex 412	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → ((𝑝 ≤ 𝑊 ∧ 𝜓) → (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋)))
11	10	reximdva 3168	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) → (∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋)))
12	2, 11	mpd 15	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ 𝐴) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
13	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓))
14		simprl 771	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝑝 ≤ 𝑊)
15		simprr 773	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝜓)
16		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → ¬ 𝑋 ≤ 𝑊)
17		nbrne2 5163	. . . . . . 7 ⊢ ((𝑝 ≤ 𝑊 ∧ ¬ 𝑋 ≤ 𝑊) → 𝑝 ≠ 𝑋)
18	14, 16, 17	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → 𝑝 ≠ 𝑋)
19	14, 15, 18	3jca 1129	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ≤ 𝑊 ∧ 𝜓)) → (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
20	19	ex 412	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) → ((𝑝 ≤ 𝑊 ∧ 𝜓) → (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋)))
21	20	reximdv 3170	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) → (∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋)))
22	13, 21	mpd 15	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ≤ 𝑊) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
23		lhpexle1lem.2	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑋 ≤ 𝑊)) → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))
24	12, 22, 23	pm2.61dda 815	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ≤ 𝑊 ∧ 𝜓 ∧ 𝑝 ≠ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   class class class wbr 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144

This theorem is referenced by:  lhpexle1  40010  lhpexle2  40012  lhpexle3  40014