Theorem 3exbidv 1958

Description: Formula-building rule for three existential quantifiers (deduction form). (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 1954	. 2 ⊢ (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
3	2	2exbidv 1957	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  ceqsex6v  3504  euotd  5490  oprabidw  7447  oprabid  7448  0mpo0  7499  eloprabga  7525  eloprabi  8065  bnj981  35492  fundcmpsurbijinj  48375  eloprab1st2nd  49861