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Theorem rabxfr 4391
Description: Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.)
Hypotheses
Ref Expression
rabxfr.1 𝑦𝐵
rabxfr.2 𝑦𝐶
rabxfr.3 (𝑦𝐷𝐴𝐷)
rabxfr.4 (𝑥 = 𝐴 → (𝜑𝜓))
rabxfr.5 (𝑦 = 𝐵𝐴 = 𝐶)
Assertion
Ref Expression
rabxfr (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐷   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rabxfr
StepHypRef Expression
1 tru 1335 . 2
2 rabxfr.1 . . 3 𝑦𝐵
3 rabxfr.2 . . 3 𝑦𝐶
4 rabxfr.3 . . . 4 (𝑦𝐷𝐴𝐷)
54adantl 275 . . 3 ((⊤ ∧ 𝑦𝐷) → 𝐴𝐷)
6 rabxfr.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
7 rabxfr.5 . . 3 (𝑦 = 𝐵𝐴 = 𝐶)
82, 3, 5, 6, 7rabxfrd 4390 . 2 ((⊤ ∧ 𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
91, 8mpan 420 1 (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wtru 1332  wcel 1480  wnfc 2268  {crab 2420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688
This theorem is referenced by: (None)
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