Theorem qimeqallb-P7 976

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

Hypothesis

Ref	Expression
qimeqallb-P7.1	⊢ Ⅎ𝑥𝜓

Assertion

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

Proof of Theorem qimeqallb-P7

Step	Hyp	Ref	Expression
1		axL4ex-P7 946	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2		qimeqallb-P7.1	. . . . . 6 ⊢ Ⅎ𝑥𝜓
3	2	nfrexgen-P7.CL 932	. . . . 5 ⊢ (∃𝑥𝜓 → 𝜓)
4	2	nfrgen-P7.CL 930	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
5	3, 4	syl-P3.24.RC 260	. . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥𝜓)
6	5	rcp-NDIMP0addall 207	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜓 → ∀𝑥𝜓))
7	1, 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:  dfpsubv-P7  977  qimeqallb-P7r  1052  qimeqallb-P7r.VR  1053  qceximl-P8  1125