Theorem rmoanim 41179

Description: Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2529. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

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 462	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓 → 𝑥 = 𝑦)))
2	1	ralbii 2980	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)))
3		rmoanim.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	r19.21 2956	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
5	2, 4	bitri 264	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
6	5	exbii 1774	. 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
7		nfv 1843	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝜓)
8	7	rmo2 3526	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
9		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦𝜓
10	9	rmo2 3526	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))
11	10	imbi2i 326	. . 3 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
12		19.37v 1910	. . 3 ⊢ (∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
13	11, 12	bitr4i 267	. 2 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)))
14	6, 8, 13	3bitr4i 292	1 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1704  Ⅎwnf 1708  ∀wral 2912  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475  df-ral 2917  df-rmo 2920

This theorem is referenced by:  2reu1  41186