Theorem eupickbi 2633

Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Assertion

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

Proof of Theorem eupickbi

Step	Hyp	Ref	Expression
1		eupicka 2631	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
2	1	ex 414	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → ∀𝑥(𝜑 → 𝜓)))
3		euex 2572	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
4		exintr 1896	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
5	3, 4	syl5com 31	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
6	2, 5	impbid 211	1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540  ∃wex 1782  ∃!weu 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-mo 2535  df-eu 2564

This theorem is referenced by:  mopickr  37232  sbaniota  43194