Theorem cbvab 2320

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 1965	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜓
3		cbvab.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
4		cbvab.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	4	equcoms 1722	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
6	5	bicomd 141	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	3, 6	sbie 1805	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
8		sbequ 1854	. . . . 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 1805	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
11		df-clab 2183	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
12		df-clab 2183	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
13	10, 11, 12	3bitr4i 212	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
14	13	eqriv 2193	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  Ⅎwnf 1474  [wsb 1776   ∈ wcel 2167  {cab 2182

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189

This theorem is referenced by:  cbvabv  2321  cbvrab  2761  cbvsbc  3018  cbvrabcsf  3150  dfdmf  4859  dfrnf  4907  funfvdm2f  5626  abrexex2g  6177  abrexex2  6181