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Theorem rexlim2d 40379
Description: Inference removing two restricted quantifiers. Same as rexlimdvv 3176, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
rexlim2d.x 𝑥𝜑
rexlim2d.y 𝑦𝜑
rexlim2d.3 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
Assertion
Ref Expression
rexlim2d (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
Distinct variable groups:   𝑦,𝐴   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rexlim2d
StepHypRef Expression
1 rexlim2d.x . 2 𝑥𝜑
2 nfv 1993 . 2 𝑥𝜒
3 rexlim2d.y . . . . 5 𝑦𝜑
4 nfv 1993 . . . . 5 𝑦 𝑥𝐴
53, 4nfan 1978 . . . 4 𝑦(𝜑𝑥𝐴)
6 nfv 1993 . . . 4 𝑦𝜒
7 rexlim2d.3 . . . . 5 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
87expdimp 452 . . . 4 ((𝜑𝑥𝐴) → (𝑦𝐵 → (𝜓𝜒)))
95, 6, 8rexlimd 3165 . . 3 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓𝜒))
109ex 449 . 2 (𝜑 → (𝑥𝐴 → (∃𝑦𝐵 𝜓𝜒)))
111, 2, 10rexlimd 3165 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnf 1857  wcel 2140  wrex 3052
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 1989  ax-6 2055  ax-7 2091  ax-12 2197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-ral 3056  df-rex 3057
This theorem is referenced by:  fourierdlem48  40893
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