Theorem morex 3021

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 2621	. . . 4 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
2		exancom 1586	. . . 4 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(φ ∧ x ∈ A))
3	1, 2	bitri 240	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(φ ∧ x ∈ A))
4		nfmo1 2215	. . . . . 6 ⊢ Ⅎx∃*xφ
5		nfe1 1732	. . . . . 6 ⊢ Ⅎx∃x(φ ∧ x ∈ A)
6	4, 5	nfan 1824	. . . . 5 ⊢ Ⅎx(∃*xφ ∧ ∃x(φ ∧ x ∈ A))
7		mopick 2266	. . . . 5 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (φ → x ∈ A))
8	6, 7	alrimi 1765	. . . 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 2413	. . . . . 6 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
12	10, 11	imbi12d 311	. . . . 5 ⊢ (x = B → ((φ → x ∈ A) ↔ (ψ → B ∈ A)))
13	9, 12	spcv 2946	. . . 4 ⊢ (∀x(φ → x ∈ A) → (ψ → B ∈ A))
14	8, 13	syl 15	. . 3 ⊢ ((∃*xφ ∧ ∃x(φ ∧ x ∈ A)) → (ψ → B ∈ A))
15	3, 14	sylan2b 461	. 2 ⊢ ((∃*xφ ∧ ∃x ∈ A φ) → (ψ → B ∈ A))
16	15	ancoms 439	1 ⊢ ((∃x ∈ A φ ∧ ∃*xφ) → (ψ → B ∈ A))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃wrex 2616  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862

This theorem is referenced by: (None)