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Theorem r19.12 3321
Description: Restricted quantifier version of 19.12 2337. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2381, ax-ext 2790. (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 3141 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑))
2 nfv 1906 . . . . 5 𝑦 𝑥𝐴
3 nfra1 3216 . . . . 5 𝑦𝑦𝐵 𝜑
42, 3nfan 1891 . . . 4 𝑦(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
54nfex 2334 . . 3 𝑦𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
61, 5nfxfr 1844 . 2 𝑦𝑥𝐴𝑦𝐵 𝜑
7 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
8 rsp 3202 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
98com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
109reximdv 3270 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
117, 10sylcom 30 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴 𝜑))
126, 11ralrimi 3213 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  wral 3135  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-ral 3140  df-rex 3141
This theorem is referenced by:  iuniin  4922  ucncn  22821  ftc1a  24561  heicant  34808  rngoid  35061  rngmgmbs4  35090  intimass  39877  intimag  39879
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