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Theorem r19.12 3210
Description: Restricted quantifier version of 19.12 2330. (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 2907 . . . 4 𝑦𝐴
2 nfra1 3088 . . . 4 𝑦𝑦𝐵 𝜑
31, 2nfrex 3153 . . 3 𝑦𝑥𝐴𝑦𝐵 𝜑
4 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
53, 4ralrimi 3104 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑)
6 rsp 3076 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
76com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
87reximdv 3162 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
98ralimia 3097 . 2 (∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
105, 9syl 17 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2155  wral 3055  wrex 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061
This theorem is referenced by:  iuniin  4687  ucncn  22368  ftc1a  24091  heicant  33800  rngoid  34055  rngmgmbs4  34084  intimass  38553  intimag  38555
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