Theorem sb8ab 2259

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 1936	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2124	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4		df-clab 2124	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
5	2, 3, 4	3bitr4ri 212	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑})
6	5	eqriv 2134	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Syntax hints:   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  [wsb 1735  {cab 2123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130

This theorem is referenced by: (None)