Theorem csbconstg 3151

Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 3150 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 2384	. 2 ⊢ Ⅎ𝑥𝐵
2	1	csbconstgf 3150	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  ⦋csb 3137

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sbc 3042  df-csb 3138

This theorem is referenced by:  sbcel1g  3156  sbceq1g  3157  sbcel2g  3158  sbceq2g  3159  csbidmg  3194  sbcbr12g  4164  sbcbr1g  4165  sbcbr2g  4166  sbcrel  4835  csbcnvg  4938  csbresg  5040  sbcfung  5375  csbfv12g  5709  csbfv2g  5710  elfvmptrab  5772  csbov12g  6089  csbov1g  6090  csbov2g  6091  csbwrdg  11247