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Theorem csbeq2d 3160
 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 (φ[A / x]B = [A / x]C)

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 2420 . . . 4 (φ → (y By C))
41, 3sbcbid 3099 . . 3 (φ → ([̣A / xy B ↔ [̣A / xy C))
54abbidv 2467 . 2 (φ → {y A / xy B} = {y A / xy C})
6 df-csb 3137 . 2 [A / x]B = {y A / xy B}
7 df-csb 3137 . 2 [A / x]C = {y A / xy C}
85, 6, 73eqtr4g 2410 1 (φ[A / x]B = [A / x]C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbeq2dv  3161
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