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Theorem r19.12 3256
Description: Restricted quantifier version of 19.12 2321. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2372, ax-ext 2709. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
Assertion
Ref Expression
r19.12 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 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
StepHypRef Expression
1 df-rex 3070 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑))
2 nfv 1917 . . . . 5 𝑦 𝑥𝐴
3 nfra1 3144 . . . . 5 𝑦𝑦𝐵 𝜑
42, 3nfan 1902 . . . 4 𝑦(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
54nfex 2318 . . 3 𝑦𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
61, 5nfxfr 1855 . 2 𝑦𝑥𝐴𝑦𝐵 𝜑
7 rsp 3131 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
87com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
98reximdv 3201 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
109com12 32 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴 𝜑))
116, 10ralrimi 3141 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1782  wcel 2106  wral 3064  wrex 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-ral 3069  df-rex 3070
This theorem is referenced by:  iuniin  4938  ucncn  23435  ftc1a  25199  heicant  35809  rngoid  36057  rngmgmbs4  36086  intimass  41232  intimag  41234
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