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Theorem ssrexf 4050
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 3976 . . 3 𝑥 𝐴𝐵
4 ssel 3977 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
54anim1d 611 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
63, 5eximd 2216 . 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 296 1 (𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1779  wcel 2108  wnfc 2890  wrex 3070  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-ss 3968
This theorem is referenced by:  iunxdif3  5095  stoweidlem34  46049
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