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Theorem ss2rexv 4048
Description: Two existential quantifications restricted to a subclass. (Contributed by AV, 11-Mar-2023.)
Assertion
Ref Expression
ss2rexv (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ss2rexv
StepHypRef Expression
1 ssrexv 4046 . . 3 (𝐴𝐵 → (∃𝑦𝐴 𝜑 → ∃𝑦𝐵 𝜑))
21reximdv 3164 . 2 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑))
3 ssrexv 4046 . 2 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
42, 3syld 47 1 (𝐴𝐵 → (∃𝑥𝐴𝑦𝐴 𝜑 → ∃𝑥𝐵𝑦𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wrex 3064  wss 3943
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-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960
This theorem is referenced by:  prpair  46723
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