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Theorem crefdf 33847
Description: A formulation of crefi 33846 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 2833 . . 3 (𝐽𝐵𝐽 ∈ CovHasRef𝐴)
3 crefi.x . . . 4 𝑋 = 𝐽
43crefi 33846 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
52, 4syl3an1b 1405 . 2 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
6 elin 3967 . . . . . 6 (𝑧 ∈ (𝒫 𝐽𝐴) ↔ (𝑧 ∈ 𝒫 𝐽𝑧𝐴))
7 crefdf.p . . . . . . 7 (𝑧𝐴𝜑)
87anim2i 617 . . . . . 6 ((𝑧 ∈ 𝒫 𝐽𝑧𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
96, 8sylbi 217 . . . . 5 (𝑧 ∈ (𝒫 𝐽𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
109anim1i 615 . . . 4 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶))
11 anass 468 . . . 4 (((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1210, 11sylib 218 . . 3 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1312reximi2 3079 . 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 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  Refcref 23510  CovHasRefccref 33841
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-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-cref 33842
This theorem is referenced by:  cmpfiref  33850  ldlfcntref  33853
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