Theorem csbconstg 3154

Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 3153 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 2386	. 2 ⊢ Ⅎ𝑥𝐵
2	1	csbconstgf 3153	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  ⦋csb 3140

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sbc 3045  df-csb 3141

This theorem is referenced by:  sbcel1g  3159  sbceq1g  3160  sbcel2g  3161  sbceq2g  3162  csbidmg  3197  sbcbr12g  4167  sbcbr1g  4168  sbcbr2g  4169  sbcrel  4838  csbcnvg  4941  csbresg  5043  sbcfung  5378  csbfv12g  5712  csbfv2g  5713  elfvmptrab  5775  csbov12g  6092  csbov1g  6093  csbov2g  6094  csbwrdg  11262