Theorem cbvab 2813

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Usage of the weaker cbvabw 2812 and cbvabv 2811 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 2515	. . . 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 2506	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	sbbii 2075	. . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
7	2, 6	bitr3i 277	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8		df-clab 2714	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
9		df-clab 2714	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
11	10	eqriv 2733	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  Ⅎwnf 1782  [wsb 2063   ∈ wcel 2107  {cab 2713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728

This theorem is referenced by:  cbvrab  3478  cdeqab1  3777  cbvsbc  3822  cbvrabcsf  3943