Theorem bnj1096 35180

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1096.1	⊢ (𝜑 → ∀𝑥𝜑)
bnj1096.2	⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑))

Assertion

Ref	Expression
bnj1096	⊢ (𝜓 → ∀𝑥𝜓)

Distinct variable groups:   𝜒,𝑥   𝜏,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bnj1096

Step	Hyp	Ref	Expression
1		bnj1096.2	. 2 ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑))
2		ax-5 1939	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
3		ax-5 1939	. . 3 ⊢ (𝜃 → ∀𝑥𝜃)
4		ax-5 1939	. . 3 ⊢ (𝜏 → ∀𝑥𝜏)
5		bnj1096.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
6	2, 3, 4, 5	bnj982 35176	. 2 ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑) → ∀𝑥(𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑))
7	1, 6	hbxfrbi 1854	1 ⊢ (𝜓 → ∀𝑥𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   ∧ w-bnj17 35084

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-bnj17 35085

This theorem is used by:  bnj964  35340  bnj981  35347  bnj983  35348  bnj1093  35377  bnj1145  35390