Theorem csbief 3139

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbief.1	⊢ 𝐴 ∈ V
csbief.2	⊢ Ⅎ𝑥𝐶
csbief.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbief	⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.1	. 2 ⊢ 𝐴 ∈ V
2		csbief.2	. . . 4 ⊢ Ⅎ𝑥𝐶
3	2	a1i 9	. . 3 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶)
4		csbief.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	3, 4	csbiegf 3138	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6	1, 5	ax-mp 5	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Ⅎwnfc 2336  Vcvv 2773  ⦋csb 3094

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-3an 983  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  df-csb 3095

This theorem is referenced by:  csbie  3140  csbing  3381  csbopabg  4126  pofun  4363  csbima12g  5048  csbiotag  5269  csbriotag  5919  csbov123g  5990  eqerlem  6658  zsumdc  11739