Theorem sbcel21v 3039

Description: Class substitution into a membership relation. One direction of sbcel2gv 3038 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 2983	. 2 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐵 ∈ V)
2		sbcel2gv 3038	. . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
3	2	biimpd 144	. 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 2158  Vcvv 2749  [wsbc 2974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975

This theorem is referenced by: (None)