Theorem sb8ab 2328

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 1994	. . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3		df-clab 2193	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
4		df-clab 2193	. . 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 2203	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}

Syntax hints:   = wceq 1373  Ⅎwnf 1484  [wsb 1786   ∈ wcel 2177  {cab 2192

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199

This theorem is referenced by: (None)