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Theorem r19.12OLD 3297
Description: Obsolete version of 19.12 2321 as of 4-Nov-2024. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2371, ax-ext 2704. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
r19.12OLD (∃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 r19.12OLD
StepHypRef Expression
1 df-rex 3071 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑))
2 nfv 1918 . . . . 5 𝑦 𝑥𝐴
3 nfra1 3266 . . . . 5 𝑦𝑦𝐵 𝜑
42, 3nfan 1903 . . . 4 𝑦(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
54nfex 2318 . . 3 𝑦𝑥(𝑥𝐴 ∧ ∀𝑦𝐵 𝜑)
61, 5nfxfr 1856 . 2 𝑦𝑥𝐴𝑦𝐵 𝜑
7 ax-1 6 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴𝑦𝐵 𝜑))
8 rsp 3229 . . . . 5 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
98com12 32 . . . 4 (𝑦𝐵 → (∀𝑦𝐵 𝜑𝜑))
109reximdv 3164 . . 3 (𝑦𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐴 𝜑))
117, 10sylcom 30 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃𝑥𝐴 𝜑))
126, 11ralrimi 3239 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-ral 3062  df-rex 3071
This theorem is referenced by: (None)
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