Theorem 19.41vvvv 1782

Description: Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)

Assertion

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

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

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

Proof of Theorem 19.41vvvv

Step	Hyp	Ref	Expression
1		19.41vvv 1781	. . 3 ⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))
2	1	exbii 1493	. 2 ⊢ (∃w∃x∃y∃z(φ ∧ ψ) ↔ ∃w(∃x∃y∃zφ ∧ ψ))
3		19.41v 1779	. 2 ⊢ (∃w(∃x∃y∃zφ ∧ ψ) ↔ (∃w∃x∃y∃zφ ∧ ψ))
4	2, 3	bitri 173	1 ⊢ (∃w∃x∃y∃z(φ ∧ ψ) ↔ (∃w∃x∃y∃zφ ∧ ψ))

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)