Theorem sbralie 2540

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 2538	. . . 4 ⊢ (∀x ∈ y φ ↔ ∀z ∈ y [z / x]φ)
2	1	sbbii 1645	. . 3 ⊢ ([x / y]∀x ∈ y φ ↔ [x / y]∀z ∈ y [z / x]φ)
3		nfv 1418	. . . 4 ⊢ Ⅎy∀z ∈ x [z / x]φ
4		raleq 2499	. . . 4 ⊢ (y = x → (∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ))
5	3, 4	sbie 1671	. . 3 ⊢ ([x / y]∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ)
6	2, 5	bitri 173	. 2 ⊢ ([x / y]∀x ∈ y φ ↔ ∀z ∈ x [z / x]φ)
7		cbvralsv 2538	. . 3 ⊢ (∀z ∈ x [z / x]φ ↔ ∀y ∈ x [y / z][z / x]φ)
8		nfv 1418	. . . . . 6 ⊢ Ⅎzφ
9	8	sbco2 1836	. . . . 5 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)
10		nfv 1418	. . . . . 6 ⊢ Ⅎxψ
11		sbralie.1	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
12	10, 11	sbie 1671	. . . . 5 ⊢ ([y / x]φ ↔ ψ)
13	9, 12	bitri 173	. . . 4 ⊢ ([y / z][z / x]φ ↔ ψ)
14	13	ralbii 2324	. . 3 ⊢ (∀y ∈ x [y / z][z / x]φ ↔ ∀y ∈ x ψ)
15	7, 14	bitri 173	. 2 ⊢ (∀z ∈ x [z / x]φ ↔ ∀y ∈ x ψ)
16	6, 15	bitri 173	1 ⊢ ([x / y]∀x ∈ y φ ↔ ∀y ∈ x ψ)

Syntax hints:   → wi 4   ↔ wb 98  [wsb 1642  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305

This theorem is referenced by: (None)