Theorem sbcel1v 3849

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2372. (Revised by Wolf Lammen, 30-Apr-2023.)

Assertion

Ref	Expression
sbcel1v	⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)

Distinct variable group:   𝑥,𝐵

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1v

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3788	. 2 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 → 𝐴 ∈ V)
2		elex 3493	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		dfsbcq2 3781	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑥 ∈ 𝐵))
4		eleq1 2822	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
5		clelsb1 2861	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)
6	3, 4, 5	vtoclbg 3560	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	1, 2, 6	pm5.21nii 380	1 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 205  [wsb 2068   ∈ wcel 2107  Vcvv 3475  [wsbc 3778

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-sbc 3779

This theorem is referenced by:  tfinds2  7853  filuni  23389  gropeld  28293  grstructeld  28294  f1od2  31946  esum2dlem  33090  bnj110  33869  f1omptsnlem  36217  relowlpssretop  36245  rdgeqoa  36251  minregex  42285  cotrclrcl  42493  frege70  42684  frege72  42686  frege91  42705  sbcoreleleq  43296  onfrALTlem4  43304  sbcoreleleqVD  43620  onfrALTlem4VD  43647