Theorem csbie 2995

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)

Hypotheses

Ref	Expression
csbie.1	⊢ 𝐴 ∈ V
csbie.2	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbie	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csbie

Step	Hyp	Ref	Expression
1		csbie.1	. 2 ⊢ 𝐴 ∈ V
2		nfcv 2240	. 2 ⊢ Ⅎ𝑥𝐶
3		csbie.2	. 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	1, 2, 3	csbief 2994	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  Vcvv 2641  ⦋csb 2955

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863  df-csb 2956

This theorem is referenced by:  fsumcnv  11045  fisum0diag2  11055