Theorem sbcel1v 3062

Description: Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcel1v	⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)

Distinct variable group:   𝑥,𝐵

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem sbcel1v

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3008	. 2 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 → 𝐴 ∈ V)
2		elex 2784	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
3		dfsbcq2 3002	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑥 ∈ 𝐵))
4		eleq1 2269	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
5		clelsb1 2311	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)
6	3, 4, 5	vtoclbg 2835	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	1, 2, 6	pm5.21nii 706	1 ⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:   ↔ wb 105  [wsb 1786   ∈ wcel 2177  Vcvv 2773  [wsbc 2999

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3000

This theorem is referenced by:  f1od2  6328  gropeld  15692  grstructeld2dom  15693