Theorem eupicka 2637

Description: Version of eupick 2636 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 2591	. . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfe1 2151	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
3	1, 2	nfan 1898	. 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))
4		eupick 2636	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	3, 4	alrimi 2214	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1777  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572

This theorem is referenced by:  eupickbi  2639  frege124d  43723  sbiota1  44403