Theorem cbvrex 1332

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvral.1	⊢ (φ → ∀yφ)
cbvral.2	⊢ (ψ → ∀xψ)
cbvral.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvrex	⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)

Distinct variable group(s):   x,y,A

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
2		cbvral.1	. . . 4 ⊢ (φ → ∀yφ)
3	1, 2	hban 704	. . 3 ⊢ ((x ∈ A ∧ φ) → ∀y(x ∈ A ∧ φ))
4		ax-17 925	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
5		cbvral.2	. . . 4 ⊢ (ψ → ∀xψ)
6	4, 5	hban 704	. . 3 ⊢ ((y ∈ A ∧ ψ) → ∀x(y ∈ A ∧ ψ))
7		eleq1 1149	. . . 4 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
8		cbvral.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
9	7, 8	anbi12d 476	. . 3 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ ψ)))
10	3, 6, 9	cbvex 849	. 2 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃y(y ∈ A ∧ ψ))
11		df-rex 1206	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
12		df-rex 1206	. 2 ⊢ (∃y ∈ A ψ ↔ ∃y(y ∈ A ∧ ψ))
13	10, 11, 12	3bitr4 158	1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  cbvrexv 1334  elrnopab 2884  abrexexlem2 2911  elrnoprab 3054

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-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-rex 1206

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