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Theorem csbiedf 2881
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1 xφ
csbiedf.2 (φx𝐶)
csbiedf.3 (φA 𝑉)
csbiedf.4 ((φ x = A) → B = 𝐶)
Assertion
Ref Expression
csbiedf (φA / xB = 𝐶)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)   𝐶(x)   𝑉(x)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3 xφ
2 csbiedf.4 . . . 4 ((φ x = A) → B = 𝐶)
32ex 108 . . 3 (φ → (x = AB = 𝐶))
41, 3alrimi 1412 . 2 (φx(x = AB = 𝐶))
5 csbiedf.3 . . 3 (φA 𝑉)
6 csbiedf.2 . . 3 (φx𝐶)
7 csbiebt 2880 . . 3 ((A 𝑉 x𝐶) → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
85, 6, 7syl2anc 391 . 2 (φ → (x(x = AB = 𝐶) ↔ A / xB = 𝐶))
94, 8mpbid 135 1 (φA / xB = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  csb 2846
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-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-sbc 2759  df-csb 2847
This theorem is referenced by:  csbied  2886  csbie2t  2888
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