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]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

Proof of Theorem sbccsb2

Step	Hyp	Ref	Expression
1		sbcex 3707	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		elex 3429	. 2 ⊢ (𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3		abid 2740	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	sbcbii 3754	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5		sbcel12 4303	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})
6		csbvarg 4326	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
7	6	eleq1d 2837	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
8	5, 7	syl5bb 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 384	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

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)