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Theorem csbie2t 2888
Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2889). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbie2t.1 A V
csbie2t.2 B V
Assertion
Ref Expression
csbie2t (xy((x = A y = B) → 𝐶 = 𝐷) → A / xB / y𝐶 = 𝐷)
Distinct variable groups:   x,y,A   x,B,y   x,𝐷,y
Allowed substitution hints:   𝐶(x,y)

Proof of Theorem csbie2t
StepHypRef Expression
1 nfa1 1431 . 2 xxy((x = A y = B) → 𝐶 = 𝐷)
2 nfcvd 2176 . 2 (xy((x = A y = B) → 𝐶 = 𝐷) → x𝐷)
3 csbie2t.1 . . 3 A V
43a1i 9 . 2 (xy((x = A y = B) → 𝐶 = 𝐷) → A V)
5 nfa2 1468 . . . 4 yxy((x = A y = B) → 𝐶 = 𝐷)
6 nfv 1418 . . . 4 y x = A
75, 6nfan 1454 . . 3 y(xy((x = A y = B) → 𝐶 = 𝐷) x = A)
8 nfcvd 2176 . . 3 ((xy((x = A y = B) → 𝐶 = 𝐷) x = A) → y𝐷)
9 csbie2t.2 . . . 4 B V
109a1i 9 . . 3 ((xy((x = A y = B) → 𝐶 = 𝐷) x = A) → B V)
11 sp 1398 . . . . 5 (y((x = A y = B) → 𝐶 = 𝐷) → ((x = A y = B) → 𝐶 = 𝐷))
1211sps 1427 . . . 4 (xy((x = A y = B) → 𝐶 = 𝐷) → ((x = A y = B) → 𝐶 = 𝐷))
1312impl 362 . . 3 (((xy((x = A y = B) → 𝐶 = 𝐷) x = A) y = B) → 𝐶 = 𝐷)
147, 8, 10, 13csbiedf 2881 . 2 ((xy((x = A y = B) → 𝐶 = 𝐷) x = A) → B / y𝐶 = 𝐷)
151, 2, 4, 14csbiedf 2881 1 (xy((x = A y = B) → 𝐶 = 𝐷) → A / xB / y𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242   wcel 1390  Vcvv 2551  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:  csbie2  2889
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