Theorem eupicka 2135

Description: Version of eupick 2134 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		hbeu1 2065	. . 3 ⊢ (∃!𝑥𝜑 → ∀𝑥∃!𝑥𝜑)
2		hbe1 1519	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥∃𝑥(𝜑 ∧ 𝜓))
3	1, 2	hban 1571	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)))
4		eupick 2134	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
5	3, 4	alrimih 1493	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371  ∃wex 1516  ∃!weu 2055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059

This theorem is referenced by:  eupickbi  2137