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Theorem sbcbid 2810
 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 / x]ψ[A / x]χ))

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4 xφ
2 sbcbid.2 . . . 4 (φ → (ψχ))
31, 2abbid 2151 . . 3 (φ → {xψ} = {xχ})
43eleq2d 2104 . 2 (φ → (A {xψ} ↔ A {xχ}))
5 df-sbc 2759 . 2 ([A / x]ψA {xψ})
6 df-sbc 2759 . 2 ([A / x]χA {xχ})
74, 5, 63bitr4g 212 1 (φ → ([A / x]ψ[A / x]χ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1346   ∈ wcel 1390  {cab 2023  [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-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 This theorem is referenced by:  sbcbidv  2811  csbeq2d  2868
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