Theorem axL11ex-P7 981

Description: Existential Form of axL11-P7 980. †

Assertion

Ref	Expression
axL11ex-P7	⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem axL11ex-P7

Step	Hyp	Ref	Expression
1		ndnfrex1-P7.8 833	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	ndnfrex2-P7.10.RC 881	. 2 ⊢ Ⅎ𝑥∃𝑦∃𝑥𝜑
3		exi-P7.CL 952	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	alloverimex-P7.GENF.RC 950	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
5	2, 4	exia-P7.RC 954	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:   → wff-imp 10  ∃wff-exists 595

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:  excomm-P7  1059