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Theorem rabxfr 4202
 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 1247 . 2
2 rabxfr.1 . . 3 𝑦𝐵
3 rabxfr.2 . . 3 𝑦𝐶
4 rabxfr.3 . . . 4 (𝑦𝐷𝐴𝐷)
54adantl 262 . . 3 ((⊤ ∧ 𝑦𝐷) → 𝐴𝐷)
6 rabxfr.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
7 rabxfr.5 . . 3 (𝑦 = 𝐵𝐴 = 𝐶)
82, 3, 5, 6, 7rabxfrd 4201 . 2 ((⊤ ∧ 𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
91, 8mpan 400 1 (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ⊤wtru 1244   ∈ wcel 1393  Ⅎwnfc 2165  {crab 2310 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559 This theorem is referenced by: (None)
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