Theorem cbvralsv 2847

Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
cbvralsv	⊢ (∀x ∈ A φ ↔ ∀y ∈ A [y / x]φ)

Distinct variable groups:   x,A   y,A   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem cbvralsv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎzφ
2		nfs1v 2106	. . 3 ⊢ Ⅎx[z / x]φ
3		sbequ12 1919	. . 3 ⊢ (x = z → (φ ↔ [z / x]φ))
4	1, 2, 3	cbvral 2832	. 2 ⊢ (∀x ∈ A φ ↔ ∀z ∈ A [z / x]φ)
5		nfv 1619	. . . 4 ⊢ Ⅎyφ
6	5	nfsb 2109	. . 3 ⊢ Ⅎy[z / x]φ
7		nfv 1619	. . 3 ⊢ Ⅎz[y / x]φ
8		sbequ 2060	. . 3 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
9	6, 7, 8	cbvral 2832	. 2 ⊢ (∀z ∈ A [z / x]φ ↔ ∀y ∈ A [y / x]φ)
10	4, 9	bitri 240	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A [y / x]φ)

Syntax hints:   ↔ wb 176  [wsb 1648  ∀wral 2615

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-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  sbralie  2849  rspsbc  3125  ralxpf  4828