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Theorem crefdf 30296
Description: A formulation of crefi 30295 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 2835 . . 3 (𝐽𝐵𝐽 ∈ CovHasRef𝐴)
3 crefi.x . . . 4 𝑋 = 𝐽
43crefi 30295 . . 3 ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
52, 4syl3an1b 1522 . 2 ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
6 elin 3957 . . . . . 6 (𝑧 ∈ (𝒫 𝐽𝐴) ↔ (𝑧 ∈ 𝒫 𝐽𝑧𝐴))
7 crefdf.p . . . . . . 7 (𝑧𝐴𝜑)
87anim2i 610 . . . . . 6 ((𝑧 ∈ 𝒫 𝐽𝑧𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
96, 8sylbi 208 . . . . 5 (𝑧 ∈ (𝒫 𝐽𝐴) → (𝑧 ∈ 𝒫 𝐽𝜑))
109anim1i 608 . . . 4 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶))
11 anass 460 . . . 4 (((𝑧 ∈ 𝒫 𝐽𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1210, 11sylib 209 . . 3 ((𝑧 ∈ (𝒫 𝐽𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑𝑧Ref𝐶)))
1312reximi2 3155 . 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 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3055  cin 3730  wss 3731  𝒫 cpw 4314   cuni 4593   class class class wbr 4808  Refcref 21584  CovHasRefccref 30290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-cref 30291
This theorem is referenced by:  cmpfiref  30299  ldlfcntref  30302
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