Theorem bj-genan 37230

Description: Generalization rule on a conjunction. Forward inference associated with 19.26 1899. (Contributed by BJ, 7-Jul-2021.)

Hypothesis

Ref	Expression
bj-genr.1	⊢ (𝜑 ∧ 𝜓)

Assertion

Ref	Expression
bj-genan	⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Proof of Theorem bj-genan

Step	Hyp	Ref	Expression
1		bj-genr.1	. . . 4 ⊢ (𝜑 ∧ 𝜓)
2	1	simpli 488	. . 3 ⊢ 𝜑
3	2	ax-gen 1824	. 2 ⊢ ∀𝑥𝜑
4	1	simpri 490	. . 3 ⊢ 𝜓
5	4	ax-gen 1824	. 2 ⊢ ∀𝑥𝜓
6	3, 5	pm3.2i 475	1 ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)

This proof depends on syntax axioms:   ∧ wa 400  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824

This proof depends on definitions:  df-bi 210  df-an 401

This theorem is used by: (None)