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Theorem rabxp 4323
Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
Hypothesis
Ref Expression
rabxp.1 (x = ⟨y, z⟩ → (φψ))
Assertion
Ref Expression
rabxp {x (A × B) ∣ φ} = {⟨y, z⟩ ∣ (y A z B ψ)}
Distinct variable groups:   x,y,z,A   x,B,y,z   φ,y,z   ψ,x
Allowed substitution hints:   φ(x)   ψ(y,z)

Proof of Theorem rabxp
StepHypRef Expression
1 elxp 4305 . . . . 5 (x (A × B) ↔ yz(x = ⟨y, z (y A z B)))
21anbi1i 431 . . . 4 ((x (A × B) φ) ↔ (yz(x = ⟨y, z (y A z B)) φ))
3 19.41vv 1780 . . . 4 (yz((x = ⟨y, z (y A z B)) φ) ↔ (yz(x = ⟨y, z (y A z B)) φ))
4 anass 381 . . . . . 6 (((x = ⟨y, z (y A z B)) φ) ↔ (x = ⟨y, z ((y A z B) φ)))
5 rabxp.1 . . . . . . . . 9 (x = ⟨y, z⟩ → (φψ))
65anbi2d 437 . . . . . . . 8 (x = ⟨y, z⟩ → (((y A z B) φ) ↔ ((y A z B) ψ)))
7 df-3an 886 . . . . . . . 8 ((y A z B ψ) ↔ ((y A z B) ψ))
86, 7syl6bbr 187 . . . . . . 7 (x = ⟨y, z⟩ → (((y A z B) φ) ↔ (y A z B ψ)))
98pm5.32i 427 . . . . . 6 ((x = ⟨y, z ((y A z B) φ)) ↔ (x = ⟨y, z (y A z B ψ)))
104, 9bitri 173 . . . . 5 (((x = ⟨y, z (y A z B)) φ) ↔ (x = ⟨y, z (y A z B ψ)))
11102exbii 1494 . . . 4 (yz((x = ⟨y, z (y A z B)) φ) ↔ yz(x = ⟨y, z (y A z B ψ)))
122, 3, 113bitr2i 197 . . 3 ((x (A × B) φ) ↔ yz(x = ⟨y, z (y A z B ψ)))
1312abbii 2150 . 2 {x ∣ (x (A × B) φ)} = {xyz(x = ⟨y, z (y A z B ψ))}
14 df-rab 2309 . 2 {x (A × B) ∣ φ} = {x ∣ (x (A × B) φ)}
15 df-opab 3810 . 2 {⟨y, z⟩ ∣ (y A z B ψ)} = {xyz(x = ⟨y, z (y A z B ψ))}
1613, 14, 153eqtr4i 2067 1 {x (A × B) ∣ φ} = {⟨y, z⟩ ∣ (y A z B ψ)}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {crab 2304  cop 3370  {copab 3808   × cxp 4286
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294
This theorem is referenced by: (None)
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