Theorem 3exbidv 1629

Description: Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Hypothesis

Ref	Expression
3exbidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
3exbidv	⊢ (φ → (∃x∃y∃zψ ↔ ∃x∃y∃zχ))

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

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

Proof of Theorem 3exbidv

Step	Hyp	Ref	Expression
1		3exbidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	exbidv 1626	. 2 ⊢ (φ → (∃zψ ↔ ∃zχ))
3	2	2exbidv 1628	1 ⊢ (φ → (∃x∃y∃zψ ↔ ∃x∃y∃zχ))

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:  ceqsex6v  2900  ins2keq  4219  ins3keq  4220  opkelins2kg  4252  opkelins3kg  4253  oprabid  5551  eloprabga  5579