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Theorem sbc6g 3818
Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)
Assertion
Ref Expression
sbc6g (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbc6g
StepHypRef Expression
1 df-sbc 3789 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
2 elab6g 3669 . 2 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
31, 2bitrid 283 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wcel 2108  {cab 2714  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by:  sbc6  3819  sbciegft  3825  sbciegftOLD  3826  sbccomlem  3869  ralsnsg  4670  fz1sbc  13640  rdgeqoa  37371  pm14.122a  44441
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