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Theorem bj-vjust 33115
Description: Remove dependency on ax-13 2408 from vjust 3352 (note the absence of DV conditions). Soundness justification theorem for df-v 3353. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-vjust {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}

Proof of Theorem bj-vjust
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equid 2097 . . . . 5 𝑥 = 𝑥
21bj-sbtv 33095 . . . 4 [𝑧 / 𝑥]𝑥 = 𝑥
3 equid 2097 . . . . 5 𝑦 = 𝑦
43bj-sbtv 33095 . . . 4 [𝑧 / 𝑦]𝑦 = 𝑦
52, 42th 254 . . 3 ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6 df-clab 2758 . . 3 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7 df-clab 2758 . . 3 (𝑧 ∈ {𝑦𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
85, 6, 73bitr4i 292 . 2 (𝑧 ∈ {𝑥𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦𝑦 = 𝑦})
98eqriv 2768 1 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  [wsb 2049  wcel 2145  {cab 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764
This theorem is referenced by: (None)
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