Theorem bj-genan 31778

Description: Generalization rule on a conjunction. Forward inference associated with 19.26 1786. (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 473	. . 3 ⊢ 𝜑
3	2	ax-gen 1713	. 2 ⊢ ∀𝑥𝜑
4	1	simpri 477	. . 3 ⊢ 𝜓
5	4	ax-gen 1713	. 2 ⊢ ∀𝑥𝜓
6	3, 5	pm3.2i 470	1 ⊢ (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Syntax hints:   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by: (None)