Theorem hban 1906

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) Reduce axiom dependencies. (Revised by SN and Sky Winter, 22-Oct-2026.)

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		pm3.2 475	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	1, 2, 3	alrimdh 1896	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝜑 ∧ 𝜓)))
5	4	imp 412	1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  hb3an  1907  bnj982  35492  bnj1351  35539  bnj1352  35540  bnj1441  35553  bnj1441g  35554  copsex2b  38266  dvelimf-o  40189  ax12indalem  40205  ax12inda2ALT  40206  hbimpg  45739  hbimpgVD  46088