Theorem mob 3678

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 3475	. . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
2		nfv 1948	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐵 ∈ V
3		nfmo1 2585	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃*𝑥𝜑
4		nfv 1948	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜓
5	2, 3, 4	nf3an 1935	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)
6		nfv 1948	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 = 𝐵 ↔ 𝜒)
7	5, 6	nfim 1930	. . . . . . . 8 ⊢ Ⅎ𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))
8		moi.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
9	8	3anbi3d 1470	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)))
10		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
11	10	bibi1d 346	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ↔ 𝜒) ↔ (𝐴 = 𝐵 ↔ 𝜒)))
12	9, 11	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐵 ↔ 𝜒)) ↔ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))))
13		moi.2	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
14	13	mob2 3676	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐵 ↔ 𝜒))
15	7, 12, 14	vtoclg1f 3534	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
16	15	com12 33	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒)))
17	16	3expib 1140	. . . . 5 ⊢ (𝐵 ∈ V → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
18	1, 17	syl 18	. . . 4 ⊢ (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
19	18	com3r 88	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))))
20	19	imp 412	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
21	20	3impib 1134	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∃*wmo 2565  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  moi  3679  rmob  3841