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Theorem vjust 2738
Description: Soundness justification theorem for df-v 2739. (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 1701 . . . . 5 𝑥 = 𝑥
21sbt 1784 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 1701 . . . . 5 𝑦 = 𝑦
43sbt 1784 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 174 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2164 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2164 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 212 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2174 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  [wsb 1762  wcel 2148  {cab 2163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170
This theorem is referenced by: (None)
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