Theorem 19.27h 1452

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 1370	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.27h.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
3	2	19.3h 1445	. . 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 1241

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 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  aaanh  1478  19.27v  1779