Theorem qcexandr-P6 761

Description: Quantifier Collection Law: Existential Quantifier Right on Conjunction (non-freeness condition).

Hypothesis

Ref	Expression
qcexandr-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qcexandr-P6	⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem qcexandr-P6

Step	Hyp	Ref	Expression
1		andasim-P3.46a 356	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ ¬ (𝜑 → ¬ ∃𝑥𝜓))
2		andasim-P3.46a 356	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
3	2	subexinf-P5 608	. . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥 ¬ (𝜑 → ¬ 𝜓))
4		exnegall-P5 598	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 → ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))
5		qcexandr-P6.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
6	5	qcallimr-P6 757	. . . . . . 7 ⊢ ((𝜑 → ∀𝑥 ¬ 𝜓) ↔ ∀𝑥(𝜑 → ¬ 𝜓))
7	6	bisym-P3.33b.RC 299	. . . . . 6 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ (𝜑 → ∀𝑥 ¬ 𝜓))
8		allnegex-P5 597	. . . . . . 7 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9	8	subimr-P3.40b.RC 328	. . . . . 6 ⊢ ((𝜑 → ∀𝑥 ¬ 𝜓) ↔ (𝜑 → ¬ ∃𝑥𝜓))
10	7, 9	bitrns-P3.33c.RC 303	. . . . 5 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ (𝜑 → ¬ ∃𝑥𝜓))
11	10	subneg-P3.39.RC 324	. . . 4 ⊢ (¬ ∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 → ¬ ∃𝑥𝜓))
12	3, 4, 11	dbitrns-P4.16.RC 429	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ ∃𝑥𝜓))
13	12	bisym-P3.33b.RC 299	. 2 ⊢ (¬ (𝜑 → ¬ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))
14	1, 13	bitrns-P3.33c.RC 303	1 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by: (None)