Theorem sbcel2gv 3788

Description: Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
sbcel2gv	⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcel2gv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2827	. 2 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
2		eleq2 2827	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
3	1, 2	sbcie2g 3758	1 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  [wsbc 3716

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  sbcel21v  3789  csbvarg  4365  bnj92  32842  bnj539  32871  minregex  41141  frege77  41548  sbcoreleleq  42155  trsbc  42160  onfrALTlem5  42162  sbcoreleleqVD  42479  trsbcVD  42497  onfrALTlem5VD  42505