Theorem cbvab 2815

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Usage of the weaker cbvabw 2813 and cbvabv 2812 are preferred. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvab.1	⊢ Ⅎ𝑦𝜑
cbvab.2	⊢ Ⅎ𝑥𝜓
cbvab.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvab	⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Proof of Theorem cbvab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvab.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	sbco2 2516	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
3		cbvab.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
4		cbvab.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbie 2507	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	sbbii 2082	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
7	2, 6	bitr3i 276	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8		df-clab 2717	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
9		df-clab 2717	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
10	7, 8, 9	3bitr4i 302	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
11	10	eqriv 2736	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541  Ⅎwnf 1789  [wsb 2070   ∈ wcel 2109  {cab 2716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-13 2373  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731

This theorem is referenced by:  cbvrab  3423  cdeqab1  3710  cbvsbc  3755  cbvrabcsf  3884