Theorem mo2icl 3016

Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)

Assertion

Ref	Expression
mo2icl	⊢ (∀x(φ → x = A) → ∃*xφ)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem mo2icl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . . . . 6 ⊢ (y = A → (x = y ↔ x = A))
2	1	imbi2d 307	. . . . 5 ⊢ (y = A → ((φ → x = y) ↔ (φ → x = A)))
3	2	albidv 1625	. . . 4 ⊢ (y = A → (∀x(φ → x = y) ↔ ∀x(φ → x = A)))
4	3	imbi1d 308	. . 3 ⊢ (y = A → ((∀x(φ → x = y) → ∃*xφ) ↔ (∀x(φ → x = A) → ∃*xφ)))
5		19.8a 1756	. . . 4 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6		nfv 1619	. . . . 5 ⊢ Ⅎyφ
7	6	mo2 2233	. . . 4 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
8	5, 7	sylibr 203	. . 3 ⊢ (∀x(φ → x = y) → ∃*xφ)
9	4, 8	vtoclg 2915	. 2 ⊢ (A ∈ V → (∀x(φ → x = A) → ∃*xφ))
10		vex 2863	. . . . . . 7 ⊢ x ∈ V
11		eleq1 2413	. . . . . . 7 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
12	10, 11	mpbii 202	. . . . . 6 ⊢ (x = A → A ∈ V)
13	12	imim2i 13	. . . . 5 ⊢ ((φ → x = A) → (φ → A ∈ V))
14	13	con3rr3 128	. . . 4 ⊢ (¬ A ∈ V → ((φ → x = A) → ¬ φ))
15	14	alimdv 1621	. . 3 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∀x ¬ φ))
16		alnex 1543	. . . 4 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
17		exmo 2249	. . . . 5 ⊢ (∃xφ ∨ ∃*xφ)
18	17	ori 364	. . . 4 ⊢ (¬ ∃xφ → ∃*xφ)
19	16, 18	sylbi 187	. . 3 ⊢ (∀x ¬ φ → ∃*xφ)
20	15, 19	syl6 29	. 2 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∃*xφ))
21	9, 20	pm2.61i 156	1 ⊢ (∀x(φ → x = A) → ∃*xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2860

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-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 2479  df-v 2862

This theorem is referenced by: (None)