Theorem sbcel21v 3019

Description: Class substitution into a membership relation. One direction of sbcel2gv 3018 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcel21v	⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem sbcel21v

Step	Hyp	Ref	Expression
1		sbcex 2963	. 2 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐵 ∈ V)
2		sbcel2gv 3018	. . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
3	2	biimpd 143	. 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵))
4	1, 3	mpcom 36	1 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 2730  [wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by: (None)