Theorem rmoanim 3823

Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2682. (Contributed by Alexander van der Vekens, 25-Jun-2017.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 24-Aug-2023.)

Hypothesis

Ref	Expression
rmoanim.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
rmoanim	⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoanim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		impexp 454	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓 → 𝑥 = 𝑦)))
2	1	ralbii 3133	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)))
3		rmoanim.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	r19.21 3179	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
5	2, 4	bitri 278	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
6	5	exbii 1849	. 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
7		df-rmo 3114	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
8		df-mo 2598	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦))
9		impexp 454	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
10	9	albii 1821	. . . . 5 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
11		df-ral 3111	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)))
12	10, 11	bitr4i 281	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
13	12	exbii 1849	. . 3 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
14	7, 8, 13	3bitri 300	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
15		df-rmo 3114	. . . . 5 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
16		df-mo 2598	. . . . 5 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝑦))
17		impexp 454	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜓 → 𝑥 = 𝑦)))
18	17	albii 1821	. . . . . . 7 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝑥 = 𝑦)))
19		df-ral 3111	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝑥 = 𝑦)))
20	18, 19	bitr4i 281	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
21	20	exbii 1849	. . . . 5 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
22	15, 16, 21	3bitri 300	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
23	22	imbi2i 339	. . 3 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
24		19.37v 1998	. . 3 ⊢ (∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
25	23, 24	bitr4i 281	. 2 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
26	6, 14, 25	3bitr4i 306	1 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111  ∃*wmo 2596  ∀wral 3106  ∃*wrmo 3109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2598  df-ral 3111  df-rmo 3114

This theorem is referenced by:  2reu1  3826