Theorem mopickr 38738

Description: "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2640) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
mopickr	⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))

Proof of Theorem mopickr

Step	Hyp	Ref	Expression
1		exancom 1868	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		moeu2 38737	. . . 4 ⊢ (∃*𝑥𝜓 ↔ (¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓))
3		19.8a 2193	. . . . . . . 8 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	con3i 154	. . . . . . 7 ⊢ (¬ ∃𝑥𝜓 → ¬ 𝜓)
5		pm2.21 123	. . . . . . 7 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
6	4, 5	syl 17	. . . . . 6 ⊢ (¬ ∃𝑥𝜓 → (𝜓 → 𝜑))
7	6	a1d 25	. . . . 5 ⊢ (¬ ∃𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
8		eupickbi 2640	. . . . . 6 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ ∀𝑥(𝜓 → 𝜑)))
9		sp 2195	. . . . . 6 ⊢ (∀𝑥(𝜓 → 𝜑) → (𝜓 → 𝜑))
10	8, 9	biimtrdi 254	. . . . 5 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
11	7, 10	jaoi 863	. . . 4 ⊢ ((¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓) → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
12	2, 11	sylbi 218	. . 3 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
13	1, 12	biimtrid 243	. 2 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜓 → 𝜑)))
14	13	imp 407	1 ⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853  ∀wal 1545  ∃wex 1786  ∃*wmo 2541  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-mo 2543  df-eu 2573

This theorem is referenced by: (None)