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Theorem vtocld 2804
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (𝜑𝐴𝑉)
vtocld.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
vtocld.3 (𝜑𝜓)
Assertion
Ref Expression
vtocld (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtocld
StepHypRef Expression
1 vtocld.1 . 2 (𝜑𝐴𝑉)
2 vtocld.2 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
3 vtocld.3 . 2 (𝜑𝜓)
4 nfv 1539 . 2 𝑥𝜑
5 nfcvd 2333 . 2 (𝜑𝑥𝐴)
6 nfvd 1540 . 2 (𝜑 → Ⅎ𝑥𝜒)
71, 2, 3, 4, 5, 6vtocldf 2803 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160
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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  funfvima3  5771  isbth  6996  frec2uzuzd  10433  setscomd  12553
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