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Theorem r19.12 3061
Description: Restricted quantifier version of 19.12 2163. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
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 nfcv 2763 . . . 4 𝑦𝐴
2 nfra1 2940 . . . 4 𝑦𝑦𝐵 𝜑
31, 2nfrex 3006 . . 3 𝑦𝑥𝐴𝑦𝐵 𝜑
4 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
53, 4ralrimi 2956 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑)
6 rsp 2928 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
76com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
87reximdv 3015 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
98ralimia 2949 . 2 (∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
105, 9syl 17 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1989  wral 2911  wrex 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917
This theorem is referenced by:  iuniin  4529  ucncn  22083  ftc1a  23794  heicant  33424  rngoid  33681  rngmgmbs4  33710  intimass  37772  intimag  37774
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