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Theorem sbccsb2 4329
 Description: Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbccsb2 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})

Proof of Theorem sbccsb2
StepHypRef Expression
1 sbcex 3707 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 elex 3429 . 2 (𝐴𝐴 / 𝑥{𝑥𝜑} → 𝐴 ∈ V)
3 abid 2740 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43sbcbii 3754 . . 3 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ [𝐴 / 𝑥]𝜑)
5 sbcel12 4303 . . . 4 ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑})
6 csbvarg 4326 . . . . 5 (𝐴 ∈ V → 𝐴 / 𝑥𝑥 = 𝐴)
76eleq1d 2837 . . . 4 (𝐴 ∈ V → (𝐴 / 𝑥𝑥𝐴 / 𝑥{𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
85, 7syl5bb 286 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥𝜑} ↔ 𝐴𝐴 / 𝑥{𝑥𝜑}))
94, 8bitr3id 288 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑}))
101, 2, 9pm5.21nii 384 1 ([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  {cab 2736  Vcvv 3410  [wsbc 3697  ⦋csb 3806 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-nul 4227 This theorem is referenced by: (None)
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