Theorem hban 1541

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Hypotheses

Ref	Expression
hb.1	⊢ (𝜑 → ∀𝑥𝜑)
hb.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
hban	⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Proof of Theorem hban

Step	Hyp	Ref	Expression
1		hb.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		hb.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3	1, 2	anim12i 336	. 2 ⊢ ((𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓))
4		19.26 1475	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
5	3, 4	sylibr 133	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1347

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 1441  ax-gen 1443

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  hbbi  1542  hb3an  1544  hbsbv  1935  mopick  2098  eupicka  2100  mopick2  2103  cleqh  2271