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Theorem crefdf 30249
Description: A formulation of crefi 30248 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Hypotheses
Ref Expression
crefi.x 𝑋 = 𝐽
crefdf.b 𝐵 = CovHasRef𝐴
crefdf.p (𝑧𝐴𝜑)
Assertion
Ref Expression
crefdf ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑧)   𝑋(𝑧)

Proof of Theorem crefdf
StepHypRef Expression
1 crefdf.b . . . 4 𝐵 = CovHasRef𝐴
21eleq2i 2841 . . 3 (𝐽𝐵𝐽 ∈ CovHasRef𝐴)
3 crefi.x . . . 4 𝑋 = 𝐽
43crefi 30248 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
52, 4syl3an1b 1508 . 2 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
6 elin 3945 . . . . . 6 (𝑧 ∈ (𝒫 𝐽𝐴) ↔ (𝑧 ∈ 𝒫 𝐽𝑧𝐴))
7 crefdf.p . . . . . . 7 (𝑧𝐴𝜑)
87anim2i 595 . . . . . 6 ((𝑧 ∈ 𝒫 𝐽𝑧𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
96, 8sylbi 207 . . . . 5 (𝑧 ∈ (𝒫 𝐽𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
109anim1i 594 . . . 4 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶))
11 anass 459 . . . 4 (((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1210, 11sylib 208 . . 3 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1312reximi2 3157 . 2 (∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶 → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
145, 13syl 17 1 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wrex 3061  cin 3720  wss 3721  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  Refcref 21525  CovHasRefccref 30243
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  ax-sep 4912
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  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-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-cref 30244
This theorem is referenced by:  cmpfiref  30252  ldlfcntref  30255
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