Theorem 2rmorex 3041

Description: Double restricted quantification with "at most one," analogous to 2moex 2275. (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 2490	. . 3 ⊢ ℲyA
2		nfre1 2671	. . 3 ⊢ Ⅎy∃y ∈ B φ
3	1, 2	nfrmo 2787	. 2 ⊢ Ⅎy∃*x ∈ A ∃y ∈ B φ
4		rspe 2676	. . . . . 6 ⊢ ((y ∈ B ∧ φ) → ∃y ∈ B φ)
5	4	ex 423	. . . . 5 ⊢ (y ∈ B → (φ → ∃y ∈ B φ))
6	5	ralrimivw 2699	. . . 4 ⊢ (y ∈ B → ∀x ∈ A (φ → ∃y ∈ B φ))
7		rmoim 3036	. . . 4 ⊢ (∀x ∈ A (φ → ∃y ∈ B φ) → (∃*x ∈ A ∃y ∈ B φ → ∃*x ∈ A φ))
8	6, 7	syl 15	. . 3 ⊢ (y ∈ B → (∃*x ∈ A ∃y ∈ B φ → ∃*x ∈ A φ))
9	8	com12 27	. 2 ⊢ (∃*x ∈ A ∃y ∈ B φ → (y ∈ B → ∃*x ∈ A φ))
10	3, 9	ralrimi 2696	1 ⊢ (∃*x ∈ A ∃y ∈ B φ → ∀y ∈ B ∃*x ∈ A φ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  ∃*wrmo 2618

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-rmo 2623

This theorem is used by: (None)