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Theorem eqsb3 2243
 Description: Substitution applied to an atomic wff (class version of equsb3 1924). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2242 . . 3 ([𝑤 / 𝑥]𝑥 = 𝐴𝑤 = 𝐴)
21sbbii 1738 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑤]𝑤 = 𝐴)
3 nfv 1508 . . 3 𝑤 𝑥 = 𝐴
43sbco2 1938 . 2 ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑥]𝑥 = 𝐴)
5 eqsb3lem 2242 . 2 ([𝑦 / 𝑤]𝑤 = 𝐴𝑦 = 𝐴)
62, 4, 53bitr3i 209 1 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1331  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132 This theorem is referenced by:  pm13.183  2822  eqsbc3  2948
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