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Theorem vjust 2552
 Description: Soundness justification theorem for df-v 2553. (Contributed by Rodolfo Medina, 27-Apr-2010.)
Assertion
Ref Expression
vjust {xx = x} = {yy = y}

Proof of Theorem vjust
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 equid 1586 . . . . 5 x = x
21sbt 1664 . . . 4 [z / x]x = x
3 equid 1586 . . . . 5 y = y
43sbt 1664 . . . 4 [z / y]y = y
52, 42th 163 . . 3 ([z / x]x = x ↔ [z / y]y = y)
6 df-clab 2024 . . 3 (z {xx = x} ↔ [z / x]x = x)
7 df-clab 2024 . . 3 (z {yy = y} ↔ [z / y]y = y)
85, 6, 73bitr4i 201 . 2 (z {xx = x} ↔ z {yy = y})
98eqriv 2034 1 {xx = x} = {yy = y}
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023 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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030 This theorem is referenced by: (None)
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