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Theorem vjust 3403
Description: Soundness justification theorem for df-v 3404. (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 2109 . . . . 5 𝑥 = 𝑥
21sbt 2580 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 2109 . . . . 5 𝑦 = 𝑦
43sbt 2580 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 255 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2804 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2804 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 294 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2814 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  [wsb 2061  wcel 2157  {cab 2803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-sb 2062  df-clab 2804  df-cleq 2810
This theorem is referenced by: (None)
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