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Theorem eqsb3 2454
 Description: Substitution applied to an atomic wff (class version of equsb3 2102). (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 2453 . . 3 ([w / x]x = Aw = A)
21sbbii 1653 . 2 ([y / w][w / x]x = A ↔ [y / w]w = A)
3 nfv 1619 . . 3 w x = A
43sbco2 2086 . 2 ([y / w][w / x]x = A ↔ [y / x]x = A)
5 eqsb3lem 2453 . 2 ([y / w]w = Ay = A)
62, 4, 53bitr3i 266 1 ([y / x]x = Ay = A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642  [wsb 1648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346 This theorem is referenced by:  pm13.183  2979  eqsbc3  3085
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