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Theorem sbcbid 3099
 Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcbid.1 xφ
sbcbid.2 (φ → (ψχ))
Assertion
Ref Expression
sbcbid (φ → ([̣A / xψ ↔ [̣A / xχ))

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4 xφ
2 sbcbid.2 . . . 4 (φ → (ψχ))
31, 2abbid 2466 . . 3 (φ → {x ψ} = {x χ})
43eleq2d 2420 . 2 (φ → (A {x ψ} ↔ A {x χ}))
5 df-sbc 3047 . 2 ([̣A / xψA {x ψ})
6 df-sbc 3047 . 2 ([̣A / xχA {x χ})
74, 5, 63bitr4g 279 1 (φ → ([̣A / xψ ↔ [̣A / xχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  [̣wsbc 3046 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-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 This theorem is referenced by:  sbcbidv  3100  csbeq2d  3160
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