Theorem sbcel1v 3804

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2401. (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 3749	. 2 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 → 𝐴 ∈ V)
2		elex 3471	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		dfsbcq2 3742	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑥 ∈ 𝐵))
4		eleq1 2848	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
5		clelsb1 2887	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)
6	3, 4, 5	vtoclbg 3519	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	1, 2, 6	pm5.21nii 381	1 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099   ∈ wcel 2145  Vcvv 3450  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3740

This theorem is used by:  tfinds2  7860  filuni  24111  gropeld  29490  grstructeld  29491  f1od2  33190  esum2dlem  34602  bnj110  35367  f1omptsnlem  38090  relowlpssretop  38118  rdgeqoa  38124  minregex  44374  cotrclrcl  44582  frege70  44773  frege72  44775  frege91  44794  sbcoreleleq  45358  onfrALTlem4  45366  sbcoreleleqVD  45681  onfrALTlem4VD  45708  rspesbcd  45760