Theorem sbralie 1439

Description: Implicit to explicit substitution that swaps variables in a quantified expression.

Hypothesis

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

Assertion

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

Distinct variable group(s):   x,y   φ,y   ψ,x

Proof of Theorem sbralie

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (φ → ∀zφ)
2		hbs1 986	. . . . 5 ⊢ ([z / x]φ → ∀x[z / x]φ)
3		sbequ12 865	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
4	1, 2, 3	cbvral 1331	. . . 4 ⊢ (∀x ∈ y φ ↔ ∀z ∈ y [z / x]φ)
5	4	bisb 855	. . 3 ⊢ ([x / y]∀x ∈ y φ ↔ [x / y]∀z ∈ y [z / x]φ)
6		ax-17 925	. . . 4 ⊢ (∀z ∈ x [z / x]φ → ∀y∀z ∈ x [z / x]φ)
7		raleq 1324	. . . 4 ⊢ (y = x → (∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ))
8	6, 7	sbie 904	. . 3 ⊢ ([x / y]∀z ∈ y [z / x]φ ↔ ∀z ∈ x [z / x]φ)
9	5, 8	bitr 151	. 2 ⊢ ([x / y]∀x ∈ y φ ↔ ∀z ∈ x [z / x]φ)
10		ax-17 925	. . 3 ⊢ ([z / x]φ → ∀y[z / x]φ)
11		hbs1 986	. . 3 ⊢ ([y / z][z / x]φ → ∀z[y / z][z / x]φ)
12		sbequ12 865	. . 3 ⊢ (z = y → ([z / x]φ ↔ [y / z][z / x]φ))
13	10, 11, 12	cbvral 1331	. 2 ⊢ (∀z ∈ x [z / x]φ ↔ ∀y ∈ x [y / z][z / x]φ)
14	1	sbco2 913	. . . 4 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)
15		ax-17 925	. . . . 5 ⊢ (ψ → ∀xψ)
16		sbralie.1	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
17	15, 16	sbie 904	. . . 4 ⊢ ([y / x]φ ↔ ψ)
18	14, 17	bitr 151	. . 3 ⊢ ([y / z][z / x]φ ↔ ψ)
19	18	biral 1223	. 2 ⊢ (∀y ∈ x [y / z][z / x]φ ↔ ∀y ∈ x ψ)
20	9, 13, 19	3bitr 155	1 ⊢ ([x / y]∀x ∈ y φ ↔ ∀y ∈ x ψ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  [wsb 852  ∀wral 1201

This theorem is referenced by:  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-ral 1205

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