Theorem 3exbidv 1929

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 1925	. 2 ⊢ (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
3	2	2exbidv 1928	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  ceqsex6v  3534  euotd  5514  oprabidw  7440  oprabid  7441  0mpo0  7492  eloprabga  7516  eloprabgaOLD  7517  eloprabi  8049  bnj981  33961  fundcmpsurbijinj  46078