Theorem sbcel2gv 3496

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 2690	. 2 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
2		eleq2 2690	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
3	1, 2	sbcie2g 3469	1 ⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcel21v  3497  csbvarg  4003  bnj92  30932  bnj539  30961  frege77  38234  sbcoreleleq  38745  trsbc  38750  onfrALTlem5  38757  sbcoreleleqVD  39095  trsbcVD  39113  onfrALTlem5VD  39121