Theorem eupickb 2168

Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eupickb	⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))

Proof of Theorem eupickb

Step	Hyp	Ref	Expression
1		eupick 2166	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
2	1	3adant2 1047	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3		3simpc 1027	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)))
4		pm3.22 265	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜑))
5	4	eximi 1653	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜓 ∧ 𝜑))
6	5	anim2i 342	. . 3 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜓 ∧ 𝜑)))
7		eupick 2166	. . 3 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓 ∧ 𝜑)) → (𝜓 → 𝜑))
8	3, 6, 7	3syl 17	. 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
9	2, 8	impbid 129	1 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∃wex 1545  ∃!weu 2086

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)