Theorem 2rmorex 3693

Description: Double restricted quantification with "at most one", analogous to 2moex 2702. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2rmorex	⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2rmorex

Step	Hyp	Ref	Expression
1		nfcv 2955	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfre1 3265	. . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
3	1, 2	nfrmow 3328	. 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑
4		rmoim 3679	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
5		rspe 3263	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑)
6	5	ex 416	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
7	6	ralrimivw 3150	. . 3 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
8	4, 7	syl11 33	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑))
9	3, 8	ralrimi 3180	1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃*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-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rmo 3114

This theorem is referenced by:  2reu2  3827