Theorem r19.12 3320

Description: Restricted quantifier version of 19.12 2331. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2380, ax-ext 2711. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)

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 3077	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑))
2		nfv 1913	. . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		nfra1 3290	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑
4	2, 3	nfan 1898	. . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)
5	4	nfex 2328	. . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)
6	1, 5	nfxfr 1851	. 2 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
7		rsp 3253	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
8	7	com12 32	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑))
9	8	reximdv 3176	. . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
10	9	com12 32	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝜑))
11	6, 10	ralrimi 3263	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-ral 3068  df-rex 3077

This theorem is referenced by:  iuniin  5027  ucncn  24315  ftc1a  26098  heicant  37615  rngoid  37862  rngmgmbs4  37891  intimass  43616  intimag  43618