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Theorem csbeq2d 2868
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypotheses
Ref Expression
csbeq2d.1 xφ
csbeq2d.2 (φB = 𝐶)
Assertion
Ref Expression
csbeq2d (φA / xB = A / x𝐶)

Proof of Theorem csbeq2d
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq2d.1 . . . 4 xφ
2 csbeq2d.2 . . . . 5 (φB = 𝐶)
32eleq2d 2104 . . . 4 (φ → (y By 𝐶))
41, 3sbcbid 2810 . . 3 (φ → ([A / x]y B[A / x]y 𝐶))
54abbidv 2152 . 2 (φ → {y[A / x]y B} = {y[A / x]y 𝐶})
6 df-csb 2847 . 2 A / xB = {y[A / x]y B}
7 df-csb 2847 . 2 A / x𝐶 = {y[A / x]y 𝐶}
85, 6, 73eqtr4g 2094 1 (φA / xB = A / x𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wnf 1346   wcel 1390  {cab 2023  [wsbc 2758  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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-sbc 2759  df-csb 2847
This theorem is referenced by:  csbeq2dv  2869
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