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Theorem eqsbc3 2825
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2157. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
eqsbc3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2789 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵[𝐴 / 𝑥]𝑥 = 𝐵))
2 eqeq1 2062 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
3 sbsbc 2791 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵[𝑦 / 𝑥]𝑥 = 𝐵)
4 eqsb3 2157 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
53, 4bitr3i 179 . 2 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
61, 2, 5vtoclbg 2631 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  [wsb 1661  [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:  sbceqal  2841  eqsbc3r  2846
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