Theorem sbcel21v 3785

Description: Class substitution into a membership relation. One direction of sbcel2gv 3784 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 3721	. 2 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐵 ∈ V)
2		sbcel2gv 3784	. . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
3	2	biimpd 228	. 2 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵))
4	1, 3	mpcom 38	1 ⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sbc 3712

This theorem is referenced by: (None)