Theorem bj-nnfand 37098

Description: Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 37097, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 37097 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 37098 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-nnfand.1	⊢ (𝜑 → Ⅎ'𝑥𝜓)
bj-nnfand.2	⊢ (𝜑 → Ⅎ'𝑥𝜒)

Assertion

Ref	Expression
bj-nnfand	⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒))

Proof of Theorem bj-nnfand

Step	Hyp	Ref	Expression
1		19.40 1893	. . 3 ⊢ (∃𝑥(𝜓 ∧ 𝜒) → (∃𝑥𝜓 ∧ ∃𝑥𝜒))
2		bj-nnfand.1	. . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜓)
3	2	bj-nnfed 37075	. . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓))
4		bj-nnfand.2	. . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
5	4	bj-nnfed 37075	. . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))
6	3, 5	anim12d 615	. . 3 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃𝑥𝜒) → (𝜓 ∧ 𝜒)))
7	1, 6	syl5 34	. 2 ⊢ (𝜑 → (∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
8	2	bj-nnfad 37072	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
9	4	bj-nnfad 37072	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
10	8, 9	anim12d 615	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒)))
11		19.26 1877	. . 3 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒))
12	10, 11	imbitrrdi 253	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))
13		df-bj-nnf 37070	. 2 ⊢ (Ⅎ'𝑥(𝜓 ∧ 𝜒) ↔ ((∃𝑥(𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)) ∧ ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))))
14	7, 12, 13	sylanbrc 589	1 ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786  Ⅎ'wnnf 37069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-bj-nnf 37070

This theorem is referenced by:  bj-nnfbid  37102