Theorem sbralie 2848

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]∀x ∈ y φ ↔ ∀y ∈ x ψ)

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

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 2846	. . . 4 ⊢ (∀x ∈ y φ ↔ ∀z ∈ y [z / x]φ)
2	1	sbbii 1653	. . 3 ⊢ ([x / y]∀x ∈ y φ ↔ [x / y]∀z ∈ y [z / x]φ)
3		nfv 1619	. . . 4 ⊢ Ⅎy∀z ∈ x [z / x]φ
4		raleq 2807	. . . 4 ⊢ (y = x → (∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ))
5	3, 4	sbie 2038	. . 3 ⊢ ([x / y]∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ)
6	2, 5	bitri 240	. 2 ⊢ ([x / y]∀x ∈ y φ ↔ ∀z ∈ x [z / x]φ)
7		cbvralsv 2846	. . 3 ⊢ (∀z ∈ x [z / x]φ ↔ ∀y ∈ x [y / z][z / x]φ)
8		nfv 1619	. . . . . 6 ⊢ Ⅎzφ
9	8	sbco2 2086	. . . . 5 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)
10		nfv 1619	. . . . . 6 ⊢ Ⅎxψ
11		sbralie.1	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
12	10, 11	sbie 2038	. . . . 5 ⊢ ([y / x]φ ↔ ψ)
13	9, 12	bitri 240	. . . 4 ⊢ ([y / z][z / x]φ ↔ ψ)
14	13	ralbii 2638	. . 3 ⊢ (∀y ∈ x [y / z][z / x]φ ↔ ∀y ∈ x ψ)
15	7, 14	bitri 240	. 2 ⊢ (∀z ∈ x [z / x]φ ↔ ∀y ∈ x ψ)
16	6, 15	bitri 240	1 ⊢ ([x / y]∀x ∈ y φ ↔ ∀y ∈ x ψ)

Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648  ∀wral 2614

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ral 2619

This theorem is referenced by: (None)