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Theorem eqsb3 2138
Description: Substitution applied to an atomic wff (class version of equsb3 1822). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([x / y]y = Ax = A)
Distinct variable group:   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eqsb3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2137 . . 3 ([w / y]y = Aw = A)
21sbbii 1645 . 2 ([x / w][w / y]y = A ↔ [x / w]w = A)
3 nfv 1418 . . 3 w y = A
43sbco2 1836 . 2 ([x / w][w / y]y = A ↔ [x / y]y = A)
5 eqsb3lem 2137 . 2 ([x / w]w = Ax = A)
62, 4, 53bitr3i 199 1 ([x / y]y = Ax = A)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030
This theorem is referenced by:  pm13.183  2675  eqsbc3  2796
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