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Theorem sbc2iegf 2822
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2iegf.1 xψ
sbc2iegf.2 yψ
sbc2iegf.3 x B 𝑊
sbc2iegf.4 ((x = A y = B) → (φψ))
Assertion
Ref Expression
sbc2iegf ((A 𝑉 B 𝑊) → ([A / x][B / y]φψ))
Distinct variable groups:   x,y,A   y,B   x,𝑉   y,𝑊
Allowed substitution hints:   φ(x,y)   ψ(x,y)   B(x)   𝑉(y)   𝑊(x)

Proof of Theorem sbc2iegf
StepHypRef Expression
1 simpl 102 . 2 ((A 𝑉 B 𝑊) → A 𝑉)
2 simpl 102 . . . 4 ((B 𝑊 x = A) → B 𝑊)
3 sbc2iegf.4 . . . . 5 ((x = A y = B) → (φψ))
43adantll 445 . . . 4 (((B 𝑊 x = A) y = B) → (φψ))
5 nfv 1418 . . . 4 y(B 𝑊 x = A)
6 sbc2iegf.2 . . . . 5 yψ
76a1i 9 . . . 4 ((B 𝑊 x = A) → Ⅎyψ)
82, 4, 5, 7sbciedf 2792 . . 3 ((B 𝑊 x = A) → ([B / y]φψ))
98adantll 445 . 2 (((A 𝑉 B 𝑊) x = A) → ([B / y]φψ))
10 nfv 1418 . . 3 x A 𝑉
11 sbc2iegf.3 . . 3 x B 𝑊
1210, 11nfan 1454 . 2 x(A 𝑉 B 𝑊)
13 sbc2iegf.1 . . 3 xψ
1413a1i 9 . 2 ((A 𝑉 B 𝑊) → Ⅎxψ)
151, 9, 12, 14sbciedf 2792 1 ((A 𝑉 B 𝑊) → ([A / x][B / y]φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  [wsbc 2758
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
This theorem is referenced by:  sbc2ie  2823  opelopabaf  4000
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