Theorem alli-P7 947

Description: Simplified '∀' Introduction Law. †

For the original form, using explicit substitution, see ndalli-P7.17 842. The inference form is axGEN-P7 933.

Hypotheses

Ref	Expression
alli-P7.1	⊢ Ⅎ𝑥𝛾
alli-P7.2	⊢ (𝛾 → 𝜑)

Assertion

Ref	Expression
alli-P7	⊢ (𝛾 → ∀𝑥𝜑)

Proof of Theorem alli-P7

Step	Hyp	Ref	Expression
1		alli-P7.1	. . 3 ⊢ Ⅎ𝑥𝛾
2	1	nfrgen-P7.CL 930	. 2 ⊢ (𝛾 → ∀𝑥𝛾)
3		alli-P7.2	. . . 4 ⊢ (𝛾 → 𝜑)
4	3	axGEN-P7 933	. . 3 ⊢ ∀𝑥(𝛾 → 𝜑)
5		axL4-P7 945	. . 3 ⊢ (∀𝑥(𝛾 → 𝜑) → (∀𝑥𝛾 → ∀𝑥𝜑))
6	4, 5	rcp-NDIME0 228	. 2 ⊢ (∀𝑥𝛾 → ∀𝑥𝜑)
7	2, 6	syl-P3.24.RC 260	1 ⊢ (𝛾 → ∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  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:  alloverimex-P7.GENF  949  allnegex-P7-L2  957  dfnfreealtif-P7  964  alloverim-P7.GENF  971  qimeqallhalf-P7  975  dfpsubv-P7  977  axL11-P7  980  alli-P7r  990  alli-P7r.VR  991  allic-P7  1007  dalloverim-P7.GENF  1025  dalloverimex-P7.GENF  1036  cbvall-P7-L1  1060  nfrcond-P8  1108