Theorem bj-vjust 31974

Description: Remove dependency on ax-13 2234 from vjust 3174 (note the absence of DV conditions). Soundness justification theorem for df-v 3175. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-vjust	⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Proof of Theorem bj-vjust

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 1926	. . . . 5 ⊢ 𝑥 = 𝑥
2	1	bj-sbtv 31954	. . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥
3		equid 1926	. . . . 5 ⊢ 𝑦 = 𝑦
4	3	bj-sbtv 31954	. . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦
5	2, 4	2th 253	. . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
6		df-clab 2597	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥)
7		df-clab 2597	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦)
8	5, 6, 7	3bitr4i 291	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦})
9	8	eqriv 2607	1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}

Syntax hints:   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603

This theorem is referenced by: (None)