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Theorem ssrexf 4048
Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
ssrexf.1 𝑥𝐴
ssrexf.2 𝑥𝐵
Assertion
Ref Expression
ssrexf (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))

Proof of Theorem ssrexf
StepHypRef Expression
1 ssrexf.1 . . . 4 𝑥𝐴
2 ssrexf.2 . . . 4 𝑥𝐵
31, 2nfss 3974 . . 3 𝑥 𝐴𝐵
4 ssel 3975 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
54anim1d 611 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
63, 5eximd 2209 . 2 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 3071 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 3071 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
96, 7, 83imtr4g 295 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1781  wcel 2106  wnfc 2883  wrex 3070  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965
This theorem is referenced by:  iunxdif3  5098  stoweidlem34  44740
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