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Theorem r19.12 3310
Description: Restricted quantifier version of 19.12 2347. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2391, ax-ext 2793. (Revised by Wolf Lammen, 17-Jun-2023.)
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 3132 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑))
2 nfv 1916 . . . . 5 𝑦 𝑥𝐴
3 nfra1 3207 . . . . 5 𝑦𝑦𝐵 𝜑
42, 3nfan 1901 . . . 4 𝑦(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
54nfex 2344 . . 3 𝑦𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
61, 5nfxfr 1854 . 2 𝑦𝑥𝐴𝑦𝐵 𝜑
7 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
8 rsp 3193 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
98com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
109reximdv 3259 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
117, 10sylcom 30 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴 𝜑))
126, 11ralrimi 3204 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1781  wcel 2115  wral 3126  wrex 3127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-ral 3131  df-rex 3132
This theorem is referenced by:  iuniin  4904  ucncn  22870  ftc1a  24619  heicant  34974  rngoid  35222  rngmgmbs4  35251  intimass  40162  intimag  40164
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