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Theorem ssrexf 3644
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 3576 . . 3 𝑥 𝐴𝐵
4 ssel 3577 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
54anim1d 587 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
63, 5eximd 2083 . 2 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 2913 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 2913 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
96, 7, 83imtr4g 285 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wcel 1987  wnfc 2748  wrex 2908  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-in 3562  df-ss 3569
This theorem is referenced by:  iunxdif3  4572  stoweidlem34  39555
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