Theorem cbvab 2733

Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

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 2403	. . . 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 2396	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	sbbii 1874	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
7	2, 6	bitr3i 265	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8		df-clab 2597	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
9		df-clab 2597	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
10	7, 8, 9	3bitr4i 291	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
11	10	eqriv 2607	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnf 1699  [wsb 1867   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603

This theorem is referenced by:  cbvabv  2734  cbvrab  3171  cbvsbc  3431  cbvrabcsf  3534  rabsnifsb  4201  rabasiun  4459  dfdmf  5239  dfrnf  5285  funfv2f  6177  abrexex2g  7036  abrexex2  7040  bnj873  30248  ptrest  32578  poimirlem26  32605  poimirlem27  32606