Theorem moi 3020

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
moi	⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ (ψ ∧ χ)) → A = B)

Distinct variable groups:   x,A   x,B   χ,x   ψ,x

Allowed substitution hints:   φ(x)   C(x)   D(x)

Proof of Theorem moi

Step	Hyp	Ref	Expression
1		moi.1	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
2		moi.2	. . . . . 6 ⊢ (x = B → (φ ↔ χ))
3	1, 2	mob 3019	. . . . 5 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (A = B ↔ χ))
4	3	biimprd 214	. . . 4 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ ψ) → (χ → A = B))
5	4	3expia 1153	. . 3 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ) → (ψ → (χ → A = B)))
6	5	imp3a 420	. 2 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ) → ((ψ ∧ χ) → A = B))
7	6	3impia 1148	1 ⊢ (((A ∈ C ∧ B ∈ D) ∧ ∃*xφ ∧ (ψ ∧ χ)) → A = B)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = 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)