Theorem cbvab 2317

Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

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.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
2	1	nfsb 1962	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜓
3		cbvab.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
4		cbvab.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	equcoms 1719	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
6	5	bicomd 141	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	3, 6	sbie 1802	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
8		sbequ 1851	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜓 ↔ [𝑧 / 𝑦]𝜓))
9	7, 8	bitr3id 194	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜓))
10	2, 9	sbie 1802	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11		df-clab 2180	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
12		df-clab 2180	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
13	10, 11, 12	3bitr4i 212	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
14	13	eqriv 2190	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  Ⅎwnf 1471  [wsb 1773   ∈ wcel 2164  {cab 2179

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186

This theorem is referenced by:  cbvabv  2318  cbvrab  2758  cbvsbc  3014  cbvrabcsf  3146  dfdmf  4855  dfrnf  4903  funfvdm2f  5622  abrexex2g  6172  abrexex2  6176