Theorem sbcel2g 2865

Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Assertion

Ref	Expression
sbcel2g	⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ B ∈ ⦋A / x⦌𝐶))

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   𝐶(x)   𝑉(x)

Proof of Theorem sbcel2g

Step	Hyp	Ref	Expression
1		sbcel12g 2859	. 2 ⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ ⦋A / x⦌B ∈ ⦋A / x⦌𝐶))
2		csbconstg 2858	. . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌B = B)
3	2	eleq1d 2103	. 2 ⊢ (A ∈ 𝑉 → (⦋A / x⦌B ∈ ⦋A / x⦌𝐶 ↔ B ∈ ⦋A / x⦌𝐶))
4	1, 3	bitrd 177	1 ⊢ (A ∈ 𝑉 → ([A / x]B ∈ 𝐶 ↔ B ∈ ⦋A / x⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  [wsbc 2758  ⦋csb 2846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847

This theorem is referenced by:  csbcomg  2867  sbccsbg  2872  sbnfc2  2900  csbabg  2901  sbcssg  3324  csbunig  3579  csbxpg  4364  csbdmg  4472  csbrng  4725  bj-sels  9369