Theorem nfrexgen-P7 931

Description: Dual Form of nfrgen-P7 928. †

Hypotheses

Ref	Expression
nfrexgen-P7.1	⊢ Ⅎ𝑥𝜑
nfrexgen-P7.2	⊢ (𝛾 → ∃𝑥𝜑)

Assertion

Ref	Expression
nfrexgen-P7	⊢ (𝛾 → 𝜑)

Proof of Theorem nfrexgen-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrexgen-P7.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
2	1	psubnfrv-P7 927	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
3	2	rcp-NDBIEF0 240	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
4	3	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → 𝜑))
5		nfrexgen-P7.2	. 2 ⊢ (𝛾 → ∃𝑥𝜑)
6	4, 5	ndexew-P7.VR123of3 874	1 ⊢ (𝛾 → 𝜑)

Syntax hints:  term-obj 1   → wff-imp 10  ∃wff-exists 595  Ⅎwff-nfree 681  [wff-psub 714

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:  nfrexgen-P7.CL  932  nfrexgen-P8  1080  nfrexgen-P8.VR  1081  nfrexgen-P8.RC  1082