Theorem morex 2719

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 2306	. . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		exancom 1496	. . . 4 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(φ ∧ x ∈ A))
3	1, 2	bitri 173	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(φ ∧ x ∈ A))
4		nfmo1 1909	. . . . . 6 ⊢ Ⅎx∃*xφ
5		nfe1 1382	. . . . . 6 ⊢ Ⅎx∃x(φ ∧ x ∈ A)
6	4, 5	nfan 1454	. . . . 5 ⊢ Ⅎx(∃*xφ ∧ ∃x(φ ∧ x ∈ A))
7		mopick 1975	. . . . 5 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (φ → x ∈ A))
8	6, 7	alrimi 1412	. . . 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 2097	. . . . . 6 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
12	10, 11	imbi12d 223	. . . . 5 ⊢ (x = B → ((φ → x ∈ A) ↔ (ψ → B ∈ A)))
13	9, 12	spcv 2640	. . . 4 ⊢ (∀x(φ → x ∈ A) → (ψ → B ∈ A))
14	8, 13	syl 14	. . 3 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (ψ → B ∈ A))
15	3, 14	sylan2b 271	. 2 ⊢ ((∃*xφ ∧ ∃x ∈ A φ) → (ψ → B ∈ A))
16	15	ancoms 255	1 ⊢ ((∃x ∈ A φ ∧ ∃*xφ) → (ψ → B ∈ A))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  ∃wrex 2301  Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553

This theorem is referenced by: (None)