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Theorem rabxfrd 4167
Description: Class builder membership after substituting an expression A (containing y) for x in the class expression χ. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
rabxfrd.1 yB
rabxfrd.2 y𝐶
rabxfrd.3 ((φ y 𝐷) → A 𝐷)
rabxfrd.4 (x = A → (ψχ))
rabxfrd.5 (y = BA = 𝐶)
Assertion
Ref Expression
rabxfrd ((φ B 𝐷) → (𝐶 {x 𝐷ψ} ↔ B {y 𝐷χ}))
Distinct variable groups:   x,A   x,y,𝐷   φ,y   ψ,y   χ,x
Allowed substitution hints:   φ(x)   ψ(x)   χ(y)   A(y)   B(x,y)   𝐶(x,y)

Proof of Theorem rabxfrd
StepHypRef Expression
1 rabxfrd.3 . . . . . . . . . . 11 ((φ y 𝐷) → A 𝐷)
21ex 108 . . . . . . . . . 10 (φ → (y 𝐷A 𝐷))
3 ibibr 235 . . . . . . . . . 10 ((y 𝐷A 𝐷) ↔ (y 𝐷 → (A 𝐷y 𝐷)))
42, 3sylib 127 . . . . . . . . 9 (φ → (y 𝐷 → (A 𝐷y 𝐷)))
54imp 115 . . . . . . . 8 ((φ y 𝐷) → (A 𝐷y 𝐷))
65anbi1d 438 . . . . . . 7 ((φ y 𝐷) → ((A 𝐷 χ) ↔ (y 𝐷 χ)))
7 rabxfrd.4 . . . . . . . 8 (x = A → (ψχ))
87elrab 2692 . . . . . . 7 (A {x 𝐷ψ} ↔ (A 𝐷 χ))
9 rabid 2479 . . . . . . 7 (y {y 𝐷χ} ↔ (y 𝐷 χ))
106, 8, 93bitr4g 212 . . . . . 6 ((φ y 𝐷) → (A {x 𝐷ψ} ↔ y {y 𝐷χ}))
1110rabbidva 2542 . . . . 5 (φ → {y 𝐷A {x 𝐷ψ}} = {y 𝐷y {y 𝐷χ}})
1211eleq2d 2104 . . . 4 (φ → (B {y 𝐷A {x 𝐷ψ}} ↔ B {y 𝐷y {y 𝐷χ}}))
13 rabxfrd.1 . . . . 5 yB
14 nfcv 2175 . . . . 5 y𝐷
15 rabxfrd.2 . . . . . 6 y𝐶
1615nfel1 2185 . . . . 5 y 𝐶 {x 𝐷ψ}
17 rabxfrd.5 . . . . . 6 (y = BA = 𝐶)
1817eleq1d 2103 . . . . 5 (y = B → (A {x 𝐷ψ} ↔ 𝐶 {x 𝐷ψ}))
1913, 14, 16, 18elrabf 2690 . . . 4 (B {y 𝐷A {x 𝐷ψ}} ↔ (B 𝐷 𝐶 {x 𝐷ψ}))
20 nfrab1 2483 . . . . . 6 y{y 𝐷χ}
2113, 20nfel 2183 . . . . 5 y B {y 𝐷χ}
22 eleq1 2097 . . . . 5 (y = B → (y {y 𝐷χ} ↔ B {y 𝐷χ}))
2313, 14, 21, 22elrabf 2690 . . . 4 (B {y 𝐷y {y 𝐷χ}} ↔ (B 𝐷 B {y 𝐷χ}))
2412, 19, 233bitr3g 211 . . 3 (φ → ((B 𝐷 𝐶 {x 𝐷ψ}) ↔ (B 𝐷 B {y 𝐷χ})))
25 pm5.32 426 . . 3 ((B 𝐷 → (𝐶 {x 𝐷ψ} ↔ B {y 𝐷χ})) ↔ ((B 𝐷 𝐶 {x 𝐷ψ}) ↔ (B 𝐷 B {y 𝐷χ})))
2624, 25sylibr 137 . 2 (φ → (B 𝐷 → (𝐶 {x 𝐷ψ} ↔ B {y 𝐷χ})))
2726imp 115 1 ((φ B 𝐷) → (𝐶 {x 𝐷ψ} ↔ B {y 𝐷χ}))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   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:  rabxfr  4168
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