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Theorem r19.12 3201
Description: Restricted quantifier version of 19.12 2309. (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 2902 . . . 4 𝑦𝐴
2 nfra1 3079 . . . 4 𝑦𝑦𝐵 𝜑
31, 2nfrex 3145 . . 3 𝑦𝑥𝐴𝑦𝐵 𝜑
4 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
53, 4ralrimi 3095 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑)
6 rsp 3067 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
76com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
87reximdv 3154 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
98ralimia 3088 . 2 (∀𝑦𝐵𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
105, 9syl 17 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  wral 3050  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056
This theorem is referenced by:  iuniin  4683  ucncn  22290  ftc1a  23999  heicant  33757  rngoid  34014  rngmgmbs4  34043  intimass  38448  intimag  38450
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