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Theorem crefdf 30756
Description: A formulation of crefi 30755 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 2851 . . 3 (𝐽𝐵𝐽 ∈ CovHasRef𝐴)
3 crefi.x . . . 4 𝑋 = 𝐽
43crefi 30755 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
52, 4syl3an1b 1383 . 2 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
6 elin 4051 . . . . . 6 (𝑧 ∈ (𝒫 𝐽𝐴) ↔ (𝑧 ∈ 𝒫 𝐽𝑧𝐴))
7 crefdf.p . . . . . . 7 (𝑧𝐴𝜑)
87anim2i 607 . . . . . 6 ((𝑧 ∈ 𝒫 𝐽𝑧𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
96, 8sylbi 209 . . . . 5 (𝑧 ∈ (𝒫 𝐽𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
109anim1i 605 . . . 4 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶))
11 anass 461 . . . 4 (((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1210, 11sylib 210 . . 3 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1312reximi2 3185 . 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 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3083  cin 3822  wss 3823  𝒫 cpw 4416   cuni 4706   class class class wbr 4923  Refcref 21808  CovHasRefccref 30750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-sep 5054
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-cref 30751
This theorem is referenced by:  cmpfiref  30759  ldlfcntref  30762
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