Theorem mob2 3353

Description: Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)

Hypothesis

Ref	Expression
moi2.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
mob2	⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem mob2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1056	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → 𝜑)
2		moi2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	syl5ibcom 234	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 → 𝜓))
4		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
5	4, 2	sbhypf 3226	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
6	5	anbi2d 736	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓)))
7		eqeq2 2621	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
8	6, 7	imbi12d 333	. . . . . 6 ⊢ (𝑦 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴)))
9	8	spcgv 3266	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴)))
10		nfs1v 2425	. . . . . . 7 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
11		sbequ12 2097	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
12	10, 11	mo4f 2504	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13		sp 2041	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14	12, 13	sylbi 206	. . . . 5 ⊢ (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
15	9, 14	impel 484	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) → ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴))
16	15	expd 451	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑) → (𝜑 → (𝜓 → 𝑥 = 𝐴)))
17	16	3impia 1253	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝜓 → 𝑥 = 𝐴))
18	3, 17	impbid 201	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃*𝑥𝜑 ∧ 𝜑) → (𝑥 = 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  moi2  3354  mob  3355  rmob2  3497