Theorem 4exbidv 1630

Description: Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)

Hypothesis

Ref	Expression
4exbidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
4exbidv	⊢ (φ → (∃x∃y∃z∃wψ ↔ ∃x∃y∃z∃wχ))

Distinct variable groups:   φ,x   φ,y   φ,z   φ,w

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

Proof of Theorem 4exbidv

Step	Hyp	Ref	Expression
1		4exbidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	2exbidv 1628	. 2 ⊢ (φ → (∃z∃wψ ↔ ∃z∃wχ))
3	2	2exbidv 1628	1 ⊢ (φ → (∃x∃y∃z∃wψ ↔ ∃x∃y∃z∃wχ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∃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

This proof depends on definitions:  df-bi 177  df-ex 1542

This theorem is used by:  ceqsex8v  2901  opbrop  4842  ov3  5600