Theorem elex22VD 45474

Description: Virtual deduction proof of elex22 3485. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elex22VD	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elex22VD

Step	Hyp	Ref	Expression
1		idn1 45210	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   )
2		simpl 487	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3	1, 2	e1a 45263	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   𝐴 ∈ 𝐵   )
4		elisset 2851	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
5	3, 4	e1a 45263	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∃𝑥 𝑥 = 𝐴   )
6		idn2 45249	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7		eleq1a 2864	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	3, 6, 7	e12 45359	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐵   )
9		simpr 489	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐶)
10	1, 9	e1a 45263	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   𝐴 ∈ 𝐶   )
11		eleq1a 2864	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶))
12	10, 6, 11	e12 45359	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐶   )
13		pm3.2 474	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
14	8, 12, 13	e22 45307	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)   )
15	14	in2 45241	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
16	15	gen11 45252	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
17		exim 1861	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
18	16, 17	e1a 45263	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
19		pm2.27 43	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
20	5, 18, 19	e11 45324	. 2 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)   )
21	20	in1 45207	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-vd1 45206  df-vd2 45214

This theorem is referenced by: (None)