Theorem 19.41vvv 1947

Description: Version of 19.41 2220 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vvv	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 1946	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑦∃𝑧𝜑 ∧ 𝜓))
2	1	exbii 1842	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓))
3		19.41v 1945	. 2 ⊢ (∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
4	2, 3	bitri 275	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774

This theorem is referenced by:  19.41vvvv  1948  eloprabga  7512  eloprabgaOLD  7513  dftpos3  8230