Theorem reupick3 3541

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reupick3	⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ) ∧ x ∈ A) → (φ → ψ))

Distinct variable group:   x,A

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

Proof of Theorem reupick3

Step	Hyp	Ref	Expression
1		df-reu 2622	. . . 4 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
2		df-rex 2621	. . . . 5 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ)))
3		anass 630	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∧ ψ) ↔ (x ∈ A ∧ (φ ∧ ψ)))
4	3	exbii 1582	. . . . 5 ⊢ (∃x((x ∈ A ∧ φ) ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ)))
5	2, 4	bitr4i 243	. . . 4 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x((x ∈ A ∧ φ) ∧ ψ))
6		eupick 2267	. . . 4 ⊢ ((∃!x(x ∈ A ∧ φ) ∧ ∃x((x ∈ A ∧ φ) ∧ ψ)) → ((x ∈ A ∧ φ) → ψ))
7	1, 5, 6	syl2anb 465	. . 3 ⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ)) → ((x ∈ A ∧ φ) → ψ))
8	7	exp3a 425	. 2 ⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ)) → (x ∈ A → (φ → ψ)))
9	8	3impia 1148	1 ⊢ ((∃!x ∈ A φ ∧ ∃x ∈ A (φ ∧ ψ) ∧ x ∈ A) → (φ → ψ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2616  ∃!wreu 2617

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

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-rex 2621  df-reu 2622

This theorem is used by:  reupick2  3542