Theorem sb8ab 2159

Description: Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)

Hypothesis

Ref	Expression
sb8ab.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8ab	⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Proof of Theorem sb8ab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb8ab.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	sbco2 1839	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2027	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4		df-clab 2027	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
5	2, 3, 4	3bitr4ri 202	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
6	5	eqriv 2037	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Syntax hints:   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  [wsb 1645  {cab 2026

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033

This theorem is referenced by: (None)