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Theorem r19.12 3305
Description: Restricted quantifier version of 19.12 2314. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2365, ax-ext 2697. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
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 3065 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑))
2 nfv 1909 . . . . 5 𝑦 𝑥𝐴
3 nfra1 3275 . . . . 5 𝑦𝑦𝐵 𝜑
42, 3nfan 1894 . . . 4 𝑦(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
54nfex 2311 . . 3 𝑦𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
61, 5nfxfr 1847 . 2 𝑦𝑥𝐴𝑦𝐵 𝜑
7 rsp 3238 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
87com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
98reximdv 3164 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
109com12 32 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴 𝜑))
116, 10ralrimi 3248 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1773  wcel 2098  wral 3055  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-ral 3056  df-rex 3065
This theorem is referenced by:  iuniin  5002  ucncn  24141  ftc1a  25923  heicant  37034  rngoid  37281  rngmgmbs4  37310  intimass  42962  intimag  42964
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