Theorem ndalli-P6 822

Description: Natural Deduction Form of Universal Quantifier Introduction.

Hypotheses

Ref	Expression
ndalli-P6.1	⊢ Ⅎ𝑦𝛾
ndalli-P6.2	⊢ Ⅎ𝑦𝜑
ndalli-P6.3	⊢ (𝛾 → [𝑦 / 𝑥]𝜑)

Assertion

Ref	Expression
ndalli-P6	⊢ (𝛾 → ∀𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem ndalli-P6

Step	Hyp	Ref	Expression
1		ndalli-P6.1	. . 3 ⊢ Ⅎ𝑦𝛾
2		ndalli-P6.3	. . 3 ⊢ (𝛾 → [𝑦 / 𝑥]𝜑)
3	1, 2	allic-P6 745	. 2 ⊢ (𝛾 → ∀𝑦[𝑦 / 𝑥]𝜑)
4		ndalli-P6.2	. . . 4 ⊢ Ⅎ𝑦𝜑
5	4	cbvallpsub-P6 768	. . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
6	5	bisym-P3.33b.RC 299	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑥𝜑)
7	3, 6	subimr2-P4.RC 543	1 ⊢ (𝛾 → ∀𝑥𝜑)

Syntax hints:  term-obj 1  ∀wff-forall 8   → wff-imp 10  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:  ndalli-P7.17  842