Theorem 19.28h 1451

Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.28h.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.28h	⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))

Proof of Theorem 19.28h

Step	Hyp	Ref	Expression
1		19.26 1367	. 2 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
2		19.28h.1	. . . 4 ⊢ (φ → ∀xφ)
3	2	19.3h 1442	. . 3 ⊢ (∀xφ ↔ φ)
4	3	anbi1i 431	. 2 ⊢ ((∀xφ ∧ ∀xψ) ↔ (φ ∧ ∀xψ))
5	1, 4	bitri 173	1 ⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240

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-4 1397

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  nfan1  1453  aaanh  1475  exan  1580  19.28v  1777