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Theorem csbidmg 3024
Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
Assertion
Ref Expression
csbidmg (𝐴𝑉𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem csbidmg
StepHypRef Expression
1 elex 2669 . 2 (𝐴𝑉𝐴 ∈ V)
2 csbnest1g 3023 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐴 / 𝑥𝐵)
3 csbconstg 2985 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 = 𝐴)
43csbeq1d 2979 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
52, 4eqtrd 2148 . 2 (𝐴 ∈ V → 𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
61, 5syl 14 1 (𝐴𝑉𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  Vcvv 2658  csb 2973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974
This theorem is referenced by: (None)
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