Theorem mob 3680

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 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐵 ∈ V
3		nfmo1 2584	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃*𝑥𝜑
4		nfv 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜓
5	2, 3, 4	nf3an 1921	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)
6		nfv 1934	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 = 𝐵 ↔ 𝜒)
7	5, 6	nfim 1916	. . . . . . . 8 ⊢ Ⅎ𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))
8		moi.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
9	8	3anbi3d 1463	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓)))
10		eqeq1 2766	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
11	10	bibi1d 345	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐵 ↔ 𝜒) ↔ (𝐴 = 𝐵 ↔ 𝜒)))
12	9, 11	imbi12d 346	. . . . . . . 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 3678	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐵 ↔ 𝜒))
15	7, 12, 14	vtoclg1f 3535	. . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
16	15	com12 32	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒)))
17	16	3expib 1135	. . . . 5 ⊢ (𝐵 ∈ V → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
18	1, 17	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 ∈ 𝐶 → (𝐴 = 𝐵 ↔ 𝜒))))
19	18	com3r 87	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))))
20	19	imp 410	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒)))
21	20	3impib 1129	1 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∃*𝑥𝜑 ∧ 𝜓) → (𝐴 = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∃*wmo 2564  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456

This theorem is referenced by:  moi  3681  rmob  3843