Theorem mob 2946

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 2774	. . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
2		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
3		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐵 ∈ V
4		nfmo1 2057	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃*𝑥𝜑
5		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜓
6	3, 4, 5	nf3an 1580	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)
7		nfv 1542	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 = 𝐵 ↔ 𝜒)
8	6, 7	nfim 1586	. . . . . . . 8 ⊢ Ⅎ𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))
9		moi.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
10	9	3anbi3d 1329	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)))
11		eqeq1 2203	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
12	11	bibi1d 233	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ↔ 𝜒) ↔ (𝐴 = 𝐵 ↔ 𝜒)))
13	10, 12	imbi12d 234	. . . . . . . 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 2944	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐵 ↔ 𝜒))
16	2, 8, 13, 15	vtoclgf 2822	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
17	16	com12 30	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒)))
18	17	3expib 1208	. . . . 5 ⊢ (𝐵 ∈ V → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
19	1, 18	syl 14	. . . 4 ⊢ (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
20	19	com3r 79	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))))
21	20	imp 124	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
22	21	3impib 1203	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃*wmo 2046   ∈ wcel 2167  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  moi  2947  rmob  3082  2omotaplemst  7325