Theorem 19.27h 1449

Description: Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.27h.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.27h	⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ))

Proof of Theorem 19.27h

Step	Hyp	Ref	Expression
1		19.26 1367	. 2 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ))
2		19.27h.1	. . . 4 ⊢ (ψ → ∀xψ)
3	2	19.3h 1442	. . 3 ⊢ (∀xψ ↔ ψ)
4	3	anbi2i 430	. 2 ⊢ ((∀xφ ∧ ∀xψ) ↔ (∀xφ ∧ ψ))
5	1, 4	bitri 173	1 ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ))

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:  aaanh  1475  19.27v  1776