Theorem eupicka 2632

Description: Version of eupick 2631 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 2586	. . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑
2		nfe1 2148	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓)
3	1, 2	nfan 1897	. 2 ⊢ Ⅎ𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))
4		eupick 2631	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	3, 4	alrimi 2211	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1776  ∃!weu 2566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567

This theorem is referenced by:  eupickbi  2634  frege124d  43751  sbiota1  44430