Theorem qimeqalla-P7 1050

Description: Quantified Implication Equivalence Law ( U ↔ ( E → U ) ) (non-freeness condition a). †

Hypothesis

Ref	Expression
qimeqalla-P7.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qimeqalla-P7	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Proof of Theorem qimeqalla-P7

Step	Hyp	Ref	Expression
1		qimeqalla-P7.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2	1	nfrexgen-P7.CL 932	. . . . 5 ⊢ (∃𝑥𝜑 → 𝜑)
3	1	nfrgen-P7.CL 930	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
4	2, 3	syl-P3.24.RC 260	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
5	4	rcp-NDIMP0addall 207	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜑))
6		axL4-P7 945	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
7	5, 6	syl-P3.24 259	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
8		qimeqallhalf-P7 975	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
9	7, 8	rcp-NDBII0 239	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃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-L10 27  ax-L11 28  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  df-psub-D6.2 716

This theorem is referenced by:  qimeqalla-P7.VR  1051  qcallimr-P8  1122