Theorem r19.12 3324

Description: Restricted quantifier version of 19.12 2342. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2386, ax-ext 2793. (Revised by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
r19.12	⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

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

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

Proof of Theorem r19.12

Step	Hyp	Ref	Expression
1		df-rex 3144	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑))
2		nfv 1911	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		nfra1 3219	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
4	2, 3	nfan 1896	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)
5	4	nfex 2339	. . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)
6	1, 5	nfxfr 1849	. 2 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
7		ax-1 6	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑))
8		rsp 3205	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
9	8	com12 32	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑))
10	9	reximdv 3273	. . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
11	7, 10	sylcom 30	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝜑))
12	6, 11	ralrimi 3216	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-ral 3143  df-rex 3144

This theorem is referenced by:  iuniin  4930  ucncn  22893  ftc1a  24633  heicant  34926  rngoid  35179  rngmgmbs4  35208  intimass  39997  intimag  39999