Theorem csbconstg 3161

Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). csbconstgf 3160 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 2392	. 2 ⊢ Ⅎ𝑥𝐵
2	1	csbconstgf 3160	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ⦋csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is used by:  sbcel1g  3166  sbceq1g  3167  sbcel2g  3168  sbceq2g  3169  csbidmg  3204  sbcbr12g  4186  sbcbr1g  4187  sbcbr2g  4188  sbcrel  4861  csbcnvg  4964  csbresg  5066  sbcfung  5401  csbfv12g  5736  csbfv2g  5737  elfvmptrab  5802  csbov12g  6125  csbov1g  6126  csbov2g  6127  csbwrdg  11336