Theorem mob 3018

Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)

Hypotheses

Ref	Expression
moi.1	⊢ (x = A → (φ ↔ ψ))
moi.2	⊢ (x = B → (φ ↔ χ))

Assertion

Ref	Expression
mob	⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ))

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

Allowed substitution hints:   φ(x)   C(x)   D(x)

Proof of Theorem mob

Step	Hyp	Ref	Expression
1		elex 2867	. . . . 5 ⊢ (B ∈ D → B ∈ V)
2		nfcv 2489	. . . . . . . 8 ⊢ ℲxA
3		nfv 1619	. . . . . . . . . 10 ⊢ Ⅎx B ∈ V
4		nfmo1 2215	. . . . . . . . . 10 ⊢ Ⅎx∃*xφ
5		nfv 1619	. . . . . . . . . 10 ⊢ Ⅎxψ
6	3, 4, 5	nf3an 1827	. . . . . . . . 9 ⊢ Ⅎx(B ∈ V ∧ ∃*xφ ∧ ψ)
7		nfv 1619	. . . . . . . . 9 ⊢ Ⅎx(A = B ↔ χ)
8	6, 7	nfim 1813	. . . . . . . 8 ⊢ Ⅎx((B ∈ V ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ))
9		moi.1	. . . . . . . . . 10 ⊢ (x = A → (φ ↔ ψ))
10	9	3anbi3d 1258	. . . . . . . . 9 ⊢ (x = A → ((B ∈ V ∧ ∃*xφ ∧ φ) ↔ (B ∈ V ∧ ∃*xφ ∧ ψ)))
11		eqeq1 2359	. . . . . . . . . 10 ⊢ (x = A → (x = B ↔ A = B))
12	11	bibi1d 310	. . . . . . . . 9 ⊢ (x = A → ((x = B ↔ χ) ↔ (A = B ↔ χ)))
13	10, 12	imbi12d 311	. . . . . . . 8 ⊢ (x = A → (((B ∈ V ∧ ∃*xφ ∧ φ) → (x = B ↔ χ)) ↔ ((B ∈ V ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ))))
14		moi.2	. . . . . . . . 9 ⊢ (x = B → (φ ↔ χ))
15	14	mob2 3016	. . . . . . . 8 ⊢ ((B ∈ V ∧ ∃*xφ ∧ φ) → (x = B ↔ χ))
16	2, 8, 13, 15	vtoclgf 2913	. . . . . . 7 ⊢ (A ∈ C → ((B ∈ V ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ)))
17	16	com12 27	. . . . . 6 ⊢ ((B ∈ V ∧ ∃*xφ ∧ ψ) → (A ∈ C → (A = B ↔ χ)))
18	17	3expib 1154	. . . . 5 ⊢ (B ∈ V → ((∃*xφ ∧ ψ) → (A ∈ C → (A = B ↔ χ))))
19	1, 18	syl 15	. . . 4 ⊢ (B ∈ D → ((∃*xφ ∧ ψ) → (A ∈ C → (A = B ↔ χ))))
20	19	com3r 73	. . 3 ⊢ (A ∈ C → (B ∈ D → ((∃*xφ ∧ ψ) → (A = B ↔ χ))))
21	20	imp 418	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → ((∃*xφ ∧ ψ) → (A = B ↔ χ)))
22	21	3impib 1149	1 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2859

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-3an 936  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 2478  df-v 2861

This theorem is referenced by:  moi  3019  rmob  3134