Theorem morex 3679

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 3063	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		exancom 1863	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
3	1, 2	bitri 275	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
4		nfmo1 2558	. . . . . 6 ⊢ Ⅎ𝑥∃*𝑥𝜑
5		nfe1 2156	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)
6	4, 5	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴))
7		mopick 2626	. . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜑 → 𝑥 ∈ 𝐴))
8	6, 7	alrimi 2221	. . . 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 2825	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
12	10, 11	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝜑 → 𝑥 ∈ 𝐴) ↔ (𝜓 → 𝐵 ∈ 𝐴)))
13	9, 12	spcv 3561	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) → (𝜓 → 𝐵 ∈ 𝐴))
14	8, 13	syl 17	. . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝑥 ∈ 𝐴)) → (𝜓 → 𝐵 ∈ 𝐴))
15	3, 14	sylan2b 595	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑) → (𝜓 → 𝐵 ∈ 𝐴))
16	15	ancoms 458	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓 → 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃wrex 3062  Vcvv 3442

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3444

This theorem is referenced by: (None)