Theorem sbcel2 4372

Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcel2	⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)

Distinct variable group:   𝑥,𝐵

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

Proof of Theorem sbcel2

Step	Hyp	Ref	Expression
1		sbcel12 4365	. . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
2		csbconstg 3870	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
3	2	eleq1d 2822	. . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
4	1, 3	bitrid 283	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
5		sbcex 3752	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 → 𝐴 ∈ V)
6	5	con3i 154	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)
7		noel 4292	. . . 4 ⊢ ¬ 𝐵 ∈ ∅
8		csbprc 4363	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
9	8	eleq2d 2823	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ∅))
10	7, 9	mtbiri 327	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
11	6, 10	2falsed 376	. 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶))
12	4, 11	pm2.61i 182	1 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2114  Vcvv 3442  [wsbc 3742  ⦋csb 3851  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-nul 4288

This theorem is referenced by:  csbcom  4374  sbccsb  4390  sbnfc2  4393  csbab  4394  sbcssg  4476  csbuni  4895  csbxp  5733  csbdm  5854  issubc  17771  esum2dlem  34269  weiunlem  36676  bj-sbeq  37143  bj-sbceqgALT  37144  f1omptsnlem  37585  csbcom2fi  38373  sbcssgVD  45232  csbingVD  45233  csbunigVD  45247  disjinfi  45545  iccelpart  47787