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Theorem sbc5 3803
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.)
Assertion
Ref Expression
sbc5 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbc5
StepHypRef Expression
1 df-sbc 3776 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
2 clelab 2871 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
31, 2bitri 274 1 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  {cab 2702  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3776
This theorem is referenced by:  sbc6gOLD  3806  sbc7  3808  sbciegftOLD  3814  sbccomlem  3862  csb2  3893  rexsns  4677  sbcop1  5493  sbccom2lem  37773  pm13.192  44021  pm13.195  44024  2sbc5g  44027  iotasbc  44030  pm14.122b  44034  iotasbc5  44042  sbcpr  47030
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