Theorem morex 3684

Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
morex.1	⊢ 𝐵 ∈ V
morex.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
morex	⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem morex

Step	Hyp	Ref	Expression
1		df-rex 3092	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		exancom 1894	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
3	1, 2	bitri 278	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
4		nfmo1 2587	. . . . . 6 ⊢ Ⅎ𝑥∃*𝑥𝜑
5		nfe1 2188	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)
6	4, 5	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
7		mopick 2655	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜑 → 𝑥 ∈ 𝐴))
8	6, 7	alrimi 2252	. . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
9		morex.1	. . . . 5 ⊢ 𝐵 ∈ V
10		morex.2	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
11		eleq1 2853	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
12	10, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜓 → 𝐵 ∈ 𝐴)))
13	9, 12	spcv 3566	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (𝜓 → 𝐵 ∈ 𝐴))
14	8, 13	syl 18	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜓 → 𝐵 ∈ 𝐴))
15	3, 14	sylan2b 606	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) → (𝜓 → 𝐵 ∈ 𝐴))
16	15	ancoms 464	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2567  ∃wrex 3091  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3092  df-v 3459

This theorem is used by: (None)