Theorem sbralie 2849

Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Hypothesis

Ref	Expression
sbralie.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
sbralie	⊢ (∀x ∈ y φ ↔ [y / x]∀y ∈ x ψ)

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

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem sbralie

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2847	. . 3 ⊢ (∀y ∈ x ψ ↔ ∀z ∈ x [z / y]ψ)
2	1	sbbii 1653	. 2 ⊢ ([y / x]∀y ∈ x ψ ↔ [y / x]∀z ∈ x [z / y]ψ)
3		nfv 1619	. . 3 ⊢ Ⅎx∀z ∈ y [z / y]ψ
4		raleq 2808	. . 3 ⊢ (x = y → (∀z ∈ x [z / y]ψ ↔ ∀z ∈ y [z / y]ψ))
5	3, 4	sbie 2038	. 2 ⊢ ([y / x]∀z ∈ x [z / y]ψ ↔ ∀z ∈ y [z / y]ψ)
6		cbvralsv 2847	. . 3 ⊢ (∀z ∈ y [z / y]ψ ↔ ∀x ∈ y [x / z][z / y]ψ)
7		nfv 1619	. . . . . 6 ⊢ Ⅎzψ
8	7	sbco2 2086	. . . . 5 ⊢ ([x / z][z / y]ψ ↔ [x / y]ψ)
9		nfv 1619	. . . . . 6 ⊢ Ⅎyφ
10		sbralie.1	. . . . . . . 8 ⊢ (x = y → (φ ↔ ψ))
11	10	bicomd 192	. . . . . . 7 ⊢ (x = y → (ψ ↔ φ))
12	11	equcoms 1681	. . . . . 6 ⊢ (y = x → (ψ ↔ φ))
13	9, 12	sbie 2038	. . . . 5 ⊢ ([x / y]ψ ↔ φ)
14	8, 13	bitri 240	. . . 4 ⊢ ([x / z][z / y]ψ ↔ φ)
15	14	ralbii 2639	. . 3 ⊢ (∀x ∈ y [x / z][z / y]ψ ↔ ∀x ∈ y φ)
16	6, 15	bitri 240	. 2 ⊢ (∀z ∈ y [z / y]ψ ↔ ∀x ∈ y φ)
17	2, 5, 16	3bitrri 263	1 ⊢ (∀x ∈ y φ ↔ [y / x]∀y ∈ x ψ)

Syntax hints:   → wi 4   ↔ 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: (None)