Theorem mopickr 38905

Description: "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 124) and *14.26 (eupickbi 2670) 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 1888	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
2		moeu2 38904	. . . 4 ⊢ (∃*𝑥𝜓 ↔ (¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓))
3		19.8a 2223	. . . . . . . 8 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	con3i 155	. . . . . . 7 ⊢ (¬ ∃𝑥𝜓 → ¬ 𝜓)
5		pm2.21 124	. . . . . . 7 ⊢ (¬ 𝜓 → (𝜓 → 𝜑))
6	4, 5	syl 18	. . . . . 6 ⊢ (¬ ∃𝑥𝜓 → (𝜓 → 𝜑))
7	6	a1d 26	. . . . 5 ⊢ (¬ ∃𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
8		eupickbi 2670	. . . . . 6 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) ↔ ∀𝑥(𝜓 → 𝜑)))
9		sp 2225	. . . . . 6 ⊢ (∀𝑥(𝜓 → 𝜑) → (𝜓 → 𝜑))
10	8, 9	biimtrdi 256	. . . . 5 ⊢ (∃!𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
11	7, 10	jaoi 870	. . . 4 ⊢ ((¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓) → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
12	2, 11	sylbi 220	. . 3 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜓 ∧ 𝜑) → (𝜓 → 𝜑)))
13	1, 12	biimtrid 245	. 2 ⊢ (∃*𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜓 → 𝜑)))
14	13	imp 411	1 ⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1565  ∃wex 1806  ∃*wmo 2571  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-mo 2573  df-eu 2603

This theorem is referenced by: (None)