Theorem elex22VD 41545

Description: Virtual deduction proof of elex22 3464. (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 41280	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   )
2		simpl 486	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)
3	1, 2	e1a 41333	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   𝐴 ∈ 𝐵   )
4		elisset 3452	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
5	3, 4	e1a 41333	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∃𝑥 𝑥 = 𝐴   )
6		idn2 41319	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7		eleq1a 2885	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	3, 6, 7	e12 41430	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐵   )
9		simpr 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐶)
10	1, 9	e1a 41333	. . . . . . . 8 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   𝐴 ∈ 𝐶   )
11		eleq1a 2885	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐶 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐶))
12	10, 6, 11	e12 41430	. . . . . . 7 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 ∈ 𝐶   )
13		pm3.2 473	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
14	8, 12, 13	e22 41377	. . . . . 6 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ,   𝑥 = 𝐴   ▶   (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)   )
15	14	in2 41311	. . . . 5 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
16	15	gen11 41322	. . . 4 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
17		exim 1835	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
18	16, 17	e1a 41333	. . 3 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))   )
19		pm2.27 42	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
20	5, 18, 19	e11 41394	. 2 ⊢ (   (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)   ▶   ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)   )
21	20	in1 41277	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-vd1 41276  df-vd2 41284

This theorem is referenced by: (None)