Theorem sb8ab 2318

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 1984	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2183	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4		df-clab 2183	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
5	2, 3, 4	3bitr4ri 213	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
6	5	eqriv 2193	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Syntax hints:   = wceq 1364  Ⅎwnf 1474  [wsb 1776   ∈ wcel 2167  {cab 2182

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189

This theorem is referenced by: (None)