Theorem eupicka 2268

Description: Version of eupick 2267 with closed formulas. (Contributed by NM, 6-Sep-2008.)

Assertion

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

Proof of Theorem eupicka

Step	Hyp	Ref	Expression
1		nfeu1 2214	. . 3 ⊢ Ⅎx∃!xφ
2		nfe1 1732	. . 3 ⊢ Ⅎx∃x(φ ∧ ψ)
3	1, 2	nfan 1824	. 2 ⊢ Ⅎx(∃!xφ ∧ ∃x(φ ∧ ψ))
4		eupick 2267	. 2 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
5	3, 4	alrimi 1765	1 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → ∀x(φ → ψ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204

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

This theorem is used by:  eupickbi  2270