Theorem nfrex2-P6 744

Description: ENF Over Existential Quantifier (different variable).

See nfrex2w-P6 695 for a version that requires only FOL axioms.

Hypothesis

Ref	Expression
nfrex2-P6.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrex2-P6	⊢ Ⅎ𝑥∃𝑦𝜑

Distinct variable group:   𝑥,𝑦

Proof of Theorem nfrex2-P6

Step	Hyp	Ref	Expression
1		excomm-P6 740	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
2	1	rcp-NDBIEF0 240	. . 3 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
3		nfrex2-P6.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
4	3	nfrexgen-P6 735	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
5	4	alloverimex-P5.RC.GEN 603	. . 3 ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑦𝜑)
6	2, 5	syl-P3.24.RC 260	. 2 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦𝜑)
7	6	exgennfr-P6 736	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  ∃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

This theorem is referenced by:  splitelof-P6  778