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Theorem sbc7 2813
Description: An equivalence for class substitution in the spirit of df-clab 2043. Note that 𝑥 and 𝐴 don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbc7 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴[𝑦 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem sbc7
StepHypRef Expression
1 sbcco 2808 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
2 sbc5 2810 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴[𝑦 / 𝑥]𝜑))
31, 2bitr3i 179 1 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴[𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wex 1397  [wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2788
This theorem is referenced by: (None)
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