Theorem cbvab 2471

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 2109	. . . 4 ⊢ Ⅎx[z / y]ψ
3		cbvab.1	. . . . . 6 ⊢ Ⅎyφ
4		cbvab.3	. . . . . . . 8 ⊢ (x = y → (φ ↔ ψ))
5	4	equcoms 1681	. . . . . . 7 ⊢ (y = x → (φ ↔ ψ))
6	5	bicomd 192	. . . . . 6 ⊢ (y = x → (ψ ↔ φ))
7	3, 6	sbie 2038	. . . . 5 ⊢ ([x / y]ψ ↔ φ)
8		sbequ 2060	. . . . 5 ⊢ (x = z → ([x / y]ψ ↔ [z / y]ψ))
9	7, 8	syl5bbr 250	. . . 4 ⊢ (x = z → (φ ↔ [z / y]ψ))
10	2, 9	sbie 2038	. . 3 ⊢ ([z / x]φ ↔ [z / y]ψ)
11		df-clab 2340	. . 3 ⊢ (z ∈ {x ∣ φ} ↔ [z / x]φ)
12		df-clab 2340	. . 3 ⊢ (z ∈ {y ∣ ψ} ↔ [z / y]ψ)
13	10, 11, 12	3bitr4i 268	. 2 ⊢ (z ∈ {x ∣ φ} ↔ z ∈ {y ∣ ψ})
14	13	eqriv 2350	1 ⊢ {x ∣ φ} = {y ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by:  cbvabv  2472  cbvrab  2857  cbvsbc  3074  cbvrabcsf  3201  dfdmf  4905  dfrnf  4962  funfv2f  5377