Theorem csbconstg 3138

Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 3137 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)

Assertion

Ref	Expression
csbconstg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem csbconstg

Step	Hyp	Ref	Expression
1		nfcv 2372	. 2 ⊢ Ⅎ𝑥𝐵
2	1	csbconstgf 3137	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  ⦋csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  sbcel1g  3143  sbceq1g  3144  sbcel2g  3145  sbceq2g  3146  csbidmg  3181  sbcbr12g  4139  sbcbr1g  4140  sbcbr2g  4141  sbcrel  4805  csbcnvg  4906  csbresg  5008  sbcfung  5342  csbfv12g  5669  csbfv2g  5670  elfvmptrab  5732  csbov12g  6047  csbov1g  6048  csbov2g  6049  csbwrdg  11109