Theorem eupickb 2269

Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eupickb	⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (φ ↔ ψ))

Proof of Theorem eupickb

Step	Hyp	Ref	Expression
1		eupick 2267	. . 3 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
2	1	3adant2 974	. 2 ⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
3		3simpc 954	. . 3 ⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (∃!xψ ∧ ∃x(φ ∧ ψ)))
4		pm3.22 436	. . . . 5 ⊢ ((φ ∧ ψ) → (ψ ∧ φ))
5	4	eximi 1576	. . . 4 ⊢ (∃x(φ ∧ ψ) → ∃x(ψ ∧ φ))
6	5	anim2i 552	. . 3 ⊢ ((∃!xψ ∧ ∃x(φ ∧ ψ)) → (∃!xψ ∧ ∃x(ψ ∧ φ)))
7		eupick 2267	. . 3 ⊢ ((∃!xψ ∧ ∃x(ψ ∧ φ)) → (ψ → φ))
8	3, 6, 7	3syl 18	. 2 ⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (ψ → φ))
9	2, 8	impbid 183	1 ⊢ ((∃!xφ ∧ ∃!xψ ∧ ∃x(φ ∧ ψ)) → (φ ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541  ∃!weu 2204

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

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

This theorem is referenced by: (None)