Theorem moi2 3018

Description: Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)

Hypothesis

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

Assertion

Ref	Expression
moi2	⊢ (((A ∈ B ∧ ∃*xφ) ∧ (φ ∧ ψ)) → x = A)

Distinct variable groups:   x,A   ψ,x

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

Proof of Theorem moi2

Step	Hyp	Ref	Expression
1		moi2.1	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
2	1	mob2 3017	. . . 4 ⊢ ((A ∈ B ∧ ∃*xφ ∧ φ) → (x = A ↔ ψ))
3	2	3expa 1151	. . 3 ⊢ (((A ∈ B ∧ ∃*xφ) ∧ φ) → (x = A ↔ ψ))
4	3	biimprd 214	. 2 ⊢ (((A ∈ B ∧ ∃*xφ) ∧ φ) → (ψ → x = A))
5	4	impr 602	1 ⊢ (((A ∈ B ∧ ∃*xφ) ∧ (φ ∧ ψ)) → x = A)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205

This proof depends on 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 proof 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 2479  df-v 2862

This theorem is used by: (None)