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Theorem opelopabsbALT 3987
 Description: The law of concretion in terms of substitutions. Less general than opelopabsb 3988, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbALT (⟨z, w {⟨x, y⟩ ∣ φ} ↔ [w / y][z / x]φ)
Distinct variable groups:   x,y,z   x,w,y
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem opelopabsbALT
StepHypRef Expression
1 excom 1551 . . 3 (xy(⟨z, w⟩ = ⟨x, y φ) ↔ yx(⟨z, w⟩ = ⟨x, y φ))
2 vex 2554 . . . . . . 7 z V
3 vex 2554 . . . . . . 7 w V
42, 3opth 3965 . . . . . 6 (⟨z, w⟩ = ⟨x, y⟩ ↔ (z = x w = y))
5 equcom 1590 . . . . . . 7 (z = xx = z)
6 equcom 1590 . . . . . . 7 (w = yy = w)
75, 6anbi12ci 434 . . . . . 6 ((z = x w = y) ↔ (y = w x = z))
84, 7bitri 173 . . . . 5 (⟨z, w⟩ = ⟨x, y⟩ ↔ (y = w x = z))
98anbi1i 431 . . . 4 ((⟨z, w⟩ = ⟨x, y φ) ↔ ((y = w x = z) φ))
1092exbii 1494 . . 3 (yx(⟨z, w⟩ = ⟨x, y φ) ↔ yx((y = w x = z) φ))
111, 10bitri 173 . 2 (xy(⟨z, w⟩ = ⟨x, y φ) ↔ yx((y = w x = z) φ))
12 elopab 3986 . 2 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ xy(⟨z, w⟩ = ⟨x, y φ))
13 2sb5 1856 . 2 ([w / y][z / x]φyx((y = w x = z) φ))
1411, 12, 133bitr4i 201 1 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ [w / y][z / x]φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  [wsb 1642  ⟨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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  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 This theorem is referenced by:  inopab  4411  cnvopab  4669
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