Theorem euim 2254

Description: Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
euim	⊢ ((∃xφ ∧ ∀x(φ → ψ)) → (∃!xψ → ∃!xφ))

Proof of Theorem euim

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (∃xφ → (∃!xψ → ∃xφ))
2		euimmo 2253	. . 3 ⊢ (∀x(φ → ψ) → (∃!xψ → ∃*xφ))
3	1, 2	anim12ii 553	. 2 ⊢ ((∃xφ ∧ ∀x(φ → ψ)) → (∃!xψ → (∃xφ ∧ ∃*xφ)))
4		eu5 2242	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
5	3, 4	syl6ibr 218	1 ⊢ ((∃xφ ∧ ∀x(φ → ψ)) → (∃!xψ → ∃!xφ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)