Theorem cbvex4v 2012

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypotheses

Ref	Expression
cbvex4v.1	⊢ ((x = v ∧ y = u) → (φ ↔ ψ))
cbvex4v.2	⊢ ((z = f ∧ w = g) → (ψ ↔ χ))

Assertion

Ref	Expression
cbvex4v	⊢ (∃x∃y∃z∃wφ ↔ ∃v∃u∃f∃gχ)

Distinct variable groups:   z,w,χ   v,u,φ   x,y,ψ   f,g,ψ   w,f   z,g   w,u,x,y,z,v

Allowed substitution hints:   φ(x, y, z, w, f, g)   ψ(z, w, v, u)   χ(x, y, v, u, f, g)

Proof of Theorem cbvex4v

Step	Hyp	Ref	Expression
1		cbvex4v.1	. . . 4 ⊢ ((x = v ∧ y = u) → (φ ↔ ψ))
2	1	2exbidv 1628	. . 3 ⊢ ((x = v ∧ y = u) → (∃z∃wφ ↔ ∃z∃wψ))
3	2	cbvex2v 2007	. 2 ⊢ (∃x∃y∃z∃wφ ↔ ∃v∃u∃z∃wψ)
4		cbvex4v.2	. . . 4 ⊢ ((z = f ∧ w = g) → (ψ ↔ χ))
5	4	cbvex2v 2007	. . 3 ⊢ (∃z∃wψ ↔ ∃f∃gχ)
6	5	2exbii 1583	. 2 ⊢ (∃v∃u∃z∃wψ ↔ ∃v∃u∃f∃gχ)
7	3, 6	bitri 240	1 ⊢ (∃x∃y∃z∃wφ ↔ ∃v∃u∃f∃gχ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541

This proof depends on 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

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by: (None)