Theorem cbvrex2v 2845

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)

Hypotheses

Ref	Expression
cbvrex2v.1	⊢ (x = z → (φ ↔ χ))
cbvrex2v.2	⊢ (y = w → (χ ↔ ψ))

Assertion

Ref	Expression
cbvrex2v	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃z ∈ A ∃w ∈ B ψ)

Distinct variable groups:   x,A   z,A   w,B   x,B,y   z,B,y   χ,w   χ,x   φ,z   ψ,y

Allowed substitution hints:   φ(x,y,w)   ψ(x,z,w)   χ(y,z)   A(y,w)

Proof of Theorem cbvrex2v

Step	Hyp	Ref	Expression
1		cbvrex2v.1	. . . 4 ⊢ (x = z → (φ ↔ χ))
2	1	rexbidv 2636	. . 3 ⊢ (x = z → (∃y ∈ B φ ↔ ∃y ∈ B χ))
3	2	cbvrexv 2837	. 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃z ∈ A ∃y ∈ B χ)
4		cbvrex2v.2	. . . 4 ⊢ (y = w → (χ ↔ ψ))
5	4	cbvrexv 2837	. . 3 ⊢ (∃y ∈ B χ ↔ ∃w ∈ B ψ)
6	5	rexbii 2640	. 2 ⊢ (∃z ∈ A ∃y ∈ B χ ↔ ∃z ∈ A ∃w ∈ B ψ)
7	3, 6	bitri 240	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃z ∈ A ∃w ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wrex 2616

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  df-rex 2621

This theorem is referenced by: (None)