Theorem 19.27h 1574

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 1495	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.27h.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
3	2	19.3h 1567	. . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 457	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
5	1, 4	bitri 184	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  aaanh  1600  19.27v  1914