Theorem 3exbii 1584

Description: Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)

Hypothesis

Ref	Expression
3exbii.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
3exbii	⊢ (∃x∃y∃zφ ↔ ∃x∃y∃zψ)

Proof of Theorem 3exbii

Step	Hyp	Ref	Expression
1		3exbii.1	. . 3 ⊢ (φ ↔ ψ)
2	1	exbii 1582	. 2 ⊢ (∃zφ ↔ ∃zψ)
3	2	2exbii 1583	1 ⊢ (∃x∃y∃zφ ↔ ∃x∃y∃zψ)

This proof depends on syntax axioms:   ↔ 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

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

This theorem is used by:  eeeanv  1914  ceqsex6v  2900  oprabid  5551  dfoprab2  5559