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Theorem rabxfr 5089
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 1658 . 2
2 rabxfr.1 . . 3 𝑦𝐵
3 rabxfr.2 . . 3 𝑦𝐶
4 rabxfr.3 . . . 4 (𝑦𝐷𝐴𝐷)
54adantl 474 . . 3 ((⊤ ∧ 𝑦𝐷) → 𝐴𝐷)
6 rabxfr.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
7 rabxfr.5 . . 3 (𝑦 = 𝐵𝐴 = 𝐶)
82, 3, 5, 6, 7rabxfrd 5088 . 2 ((⊤ ∧ 𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
91, 8mpan 682 1 (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wtru 1654  wcel 2157  wnfc 2929  {crab 3094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rab 3099  df-v 3388
This theorem is referenced by: (None)
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