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Theorem rabxfr 4168
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 (y 𝐷A 𝐷)
rabxfr.4 (x = A → (φψ))
rabxfr.5 (y = BA = 𝐶)
Assertion
Ref Expression
rabxfr (B 𝐷 → (𝐶 {x 𝐷φ} ↔ B {y 𝐷ψ}))
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 1246 . 2
2 rabxfr.1 . . 3 yB
3 rabxfr.2 . . 3 y𝐶
4 rabxfr.3 . . . 4 (y 𝐷A 𝐷)
54adantl 262 . . 3 (( ⊤ y 𝐷) → A 𝐷)
6 rabxfr.4 . . 3 (x = A → (φψ))
7 rabxfr.5 . . 3 (y = BA = 𝐶)
82, 3, 5, 6, 7rabxfrd 4167 . 2 (( ⊤ B 𝐷) → (𝐶 {x 𝐷φ} ↔ B {y 𝐷ψ}))
91, 8mpan 400 1 (B 𝐷 → (𝐶 {x 𝐷φ} ↔ B {y 𝐷ψ}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wtru 1243   wcel 1390  wnfc 2162  {crab 2304
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553
This theorem is referenced by: (None)
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