Theorem sbccsb2 4384

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 3746	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		elex 3457	. 2 ⊢ (𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3		abid 2713	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	sbcbii 3793	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5		sbcel12 4358	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})
6		csbvarg 4381	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
7	6	eleq1d 2816	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
8	5, 7	bitrid 283	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
9	4, 8	bitr3id 285	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
10	1, 2, 9	pm5.21nii 378	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

Syntax hints:   ↔ wb 206   ∈ wcel 2111  {cab 2709  Vcvv 3436  [wsbc 3736  ⦋csb 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-nul 4281

This theorem is referenced by: (None)