Theorem 19.28h 1499

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 1415	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.28h.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.3h 1490	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
4	3	anbi1i 446	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
5	1, 4	bitri 182	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nfan1  1501  aaanh  1523  exan  1628  19.28v  1828