Theorem 2rmorex 3737

Description: Double restricted quantification with "at most one", analogous to 2moex 2639. (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 2898	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfre1 3267	. . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑
3	1, 2	nfrmow 3392	. 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑
4		rmoim 3723	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑))
5		rspe 3232	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑)
6	5	ex 412	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑))
7	6	ralrimivw 3136	. . 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 3240	1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ∃*wrmo 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2539  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3359

This theorem is referenced by:  2reu2  3873