Theorem mob2 3016

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 957	. . 3 ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → φ)
2		moi2.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	1, 2	syl5ibcom 211	. 2 ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → (x = A → ψ))
4		nfs1v 2106	. . . . . . . 8 ⊢ Ⅎx[y / x]φ
5		sbequ12 1919	. . . . . . . 8 ⊢ (x = y → (φ ↔ [y / x]φ))
6	4, 5	mo4f 2236	. . . . . . 7 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
7		sp 1747	. . . . . . 7 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) → ∀y((φ ∧ [y / x]φ) → x = y))
8	6, 7	sylbi 187	. . . . . 6 ⊢ (∃*xφ → ∀y((φ ∧ [y / x]φ) → x = y))
9		nfv 1619	. . . . . . . . . 10 ⊢ Ⅎxψ
10	9, 2	sbhypf 2904	. . . . . . . . 9 ⊢ (y = A → ([y / x]φ ↔ ψ))
11	10	anbi2d 684	. . . . . . . 8 ⊢ (y = A → ((φ ∧ [y / x]φ) ↔ (φ ∧ ψ)))
12		eqeq2 2362	. . . . . . . 8 ⊢ (y = A → (x = y ↔ x = A))
13	11, 12	imbi12d 311	. . . . . . 7 ⊢ (y = A → (((φ ∧ [y / x]φ) → x = y) ↔ ((φ ∧ ψ) → x = A)))
14	13	spcgv 2939	. . . . . 6 ⊢ (A ∈ B → (∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ ψ) → x = A)))
15	8, 14	syl5 28	. . . . 5 ⊢ (A ∈ B → (∃*xφ → ((φ ∧ ψ) → x = A)))
16	15	imp 418	. . . 4 ⊢ ((A ∈ B ∧ ∃*xφ) → ((φ ∧ ψ) → x = A))
17	16	exp3a 425	. . 3 ⊢ ((A ∈ B ∧ ∃*xφ) → (φ → (ψ → x = A)))
18	17	3impia 1148	. 2 ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → (ψ → x = A))
19	3, 18	impbid 183	1 ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → (x = A ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃*wmo 2205

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:  moi2  3017  mob  3018