Theorem axL11-P7 980

Description: ax-L11 28 Derived from Natural Deduction Rules. †

Assertion

Ref	Expression
axL11-P7	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem axL11-P7

Step	Hyp	Ref	Expression
1		ndnfrall1-P7.7 832	. . 3 ⊢ Ⅎ𝑦∀𝑦𝜑
2	1	ndnfrall2-P7.9.RC 880	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦𝜑
3		alle-P7.CL 942	. . 3 ⊢ (∀𝑦𝜑 → 𝜑)
4	3	alloverim-P7.GENF.RC 972	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥𝜑)
5	2, 4	alli-P7 947	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10

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:  allcomm-P7  1058