Theorem sbccsb2 4388

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 3784	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		elex 3514	. 2 ⊢ (𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} → 𝐴 ∈ V)
3		abid 2805	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	sbcbii 3831	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5		sbcel12 4362	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})
6		csbvarg 4385	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)
7	6	eleq1d 2899	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝑥 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
8	5, 7	syl5bb 285	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
9	4, 8	syl5bbr 287	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑}))
10	1, 2, 9	pm5.21nii 382	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ ⦋𝐴 / 𝑥⦌{𝑥 ∣ 𝜑})

Syntax hints:   ↔ wb 208   ∈ wcel 2114  {cab 2801  Vcvv 3496  [wsbc 3774  ⦋csb 3885

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-nul 4294

This theorem is referenced by: (None)