Theorem sbccsb2 4408

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]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

Proof of Theorem sbccsb2

Step	Hyp	Ref	Expression
1		sbcex 3763	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		elex 3484	. 2 ⊢ (𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3		abid 2751	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	sbcbii 3809	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5		sbcel12 4382	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})
6		csbvarg 4405	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
7	6	eleq1d 2854	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
8	5, 7	bitrid 286	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
9	4, 8	bitr3id 288	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
10	1, 2, 9	pm5.21nii 381	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

Syntax hints:   ↔ wb 209   ∈ wcel 2149  {cab 2747  Vcvv 3463  [wsbc 3753  ⦋csb 3861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-nul 4295

This theorem is referenced by: (None)