Theorem r19.12 3045

Description: Restricted quantifier version of 19.12 2150. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

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		nfcv 2751	. . . 4 ⊢ Ⅎ𝑦𝐴
2		nfra1 2925	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
3	1, 2	nfrex 2990	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
4		ax-1 6	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑))
5	3, 4	ralrimi 2940	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
6		rsp 2913	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
7	6	com12 32	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑))
8	7	reximdv 2999	. . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
9	8	ralimia 2934	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
10	5, 9	syl 17	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902

This theorem is referenced by:  iuniin  4467  ucncn  21899  ftc1a  23604  heicant  32614  rngoid  32871  rngmgmbs4  32900  intimass  36965  intimag  36967