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Theorem opelopabf 4002
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 3999 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
opelopabf.x xψ
opelopabf.y yχ
opelopabf.1 A V
opelopabf.2 B V
opelopabf.3 (x = A → (φψ))
opelopabf.4 (y = B → (ψχ))
Assertion
Ref Expression
opelopabf (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ)
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)

Proof of Theorem opelopabf
StepHypRef Expression
1 opelopabsb 3988 . 2 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ [A / x][B / y]φ)
2 opelopabf.1 . . 3 A V
3 nfcv 2175 . . . . 5 xB
4 opelopabf.x . . . . 5 xψ
53, 4nfsbc 2778 . . . 4 x[B / y]ψ
6 opelopabf.3 . . . . 5 (x = A → (φψ))
76sbcbidv 2811 . . . 4 (x = A → ([B / y]φ[B / y]ψ))
85, 7sbciegf 2788 . . 3 (A V → ([A / x][B / y]φ[B / y]ψ))
92, 8ax-mp 7 . 2 ([A / x][B / y]φ[B / y]ψ)
10 opelopabf.2 . . 3 B V
11 opelopabf.y . . . 4 yχ
12 opelopabf.4 . . . 4 (y = B → (ψχ))
1311, 12sbciegf 2788 . . 3 (B V → ([B / y]ψχ))
1410, 13ax-mp 7 . 2 ([B / y]ψχ)
151, 9, 143bitri 195 1 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wnf 1346   wcel 1390  Vcvv 2551  [wsbc 2758  cop 3370  {copab 3808
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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by:  pofun  4040  fmptco  5273
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