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Theorem ssrexf 3812
 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 3743 . . 3 𝑥 𝐴𝐵
4 ssel 3744 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
54anim1d 590 . . 3 (𝐴𝐵 → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
63, 5eximd 2240 . 2 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐵𝜑)))
7 df-rex 3066 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
8 df-rex 3066 . 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 382  ∃wex 1851   ∈ wcel 2144  Ⅎwnfc 2899  ∃wrex 3061   ⊆ wss 3721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-in 3728  df-ss 3735 This theorem is referenced by:  iunxdif3  4738  stoweidlem34  40762
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