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Theorem csbie2t 3180
 Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3181). (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 / x][B / y]C = 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 1788 . 2 xxy((x = A y = B) → C = D)
2 nfcvd 2490 . 2 (xy((x = A y = B) → C = D) → xD)
3 csbie2t.1 . . 3 A V
43a1i 10 . 2 (xy((x = A y = B) → C = D) → A V)
5 nfa2 1855 . . . 4 yxy((x = A y = B) → C = D)
6 nfv 1619 . . . 4 y x = A
75, 6nfan 1824 . . 3 y(xy((x = A y = B) → C = D) x = A)
8 nfcvd 2490 . . 3 ((xy((x = A y = B) → C = D) x = A) → yD)
9 csbie2t.2 . . . 4 B V
109a1i 10 . . 3 ((xy((x = A y = B) → C = D) x = A) → B V)
11 sp 1747 . . . . 5 (y((x = A y = B) → C = D) → ((x = A y = B) → C = D))
1211sps 1754 . . . 4 (xy((x = A y = B) → C = D) → ((x = A y = B) → C = D))
1312impl 603 . . 3 (((xy((x = A y = B) → C = D) x = A) y = B) → C = D)
147, 8, 10, 13csbiedf 3173 . 2 ((xy((x = A y = B) → C = D) x = A) → [B / y]C = D)
151, 2, 4, 14csbiedf 3173 1 (xy((x = A y = B) → C = D) → [A / x][B / y]C = D)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbie2  3181
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