Theorem 19.28h 1550

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 1469	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.28h.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.3h 1541	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
4	3	anbi1i 454	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
5	1, 4	bitri 183	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nfan1  1552  aaanh  1574  exan  1681  19.28v  1888