Theorem ndalle-P7.18 843

Description: Natural Deduction: '∀' Elimination Rule.

In the traditional textbook presentation '𝜑' and '[𝑡 / 𝑥]𝜑' are written as '𝜑(𝑥)' and '𝜑(𝑡)', respectively. The rule is then stated as...

'⊢ (𝛾 → ∀𝑥𝜑(𝑥))   ⇒   ⊢ (𝛾 → 𝜑(𝑡))'.

The English interpretation of this rule is as follows -- since '𝜑(𝑥)' is true for *any* '𝑥', *in particular*, it is true for '𝑡', thus we can deduce '𝜑(𝑡)'. Within a proof, the phrase "in particular" is a good clue that this rule is being used.

Hypothesis

Ref	Expression
ndalle-P7.18.1	⊢ (𝛾 → ∀𝑥𝜑)

Assertion

Ref	Expression
ndalle-P7.18	⊢ (𝛾 → [𝑡 / 𝑥]𝜑)

Proof of Theorem ndalle-P7.18

Step	Hyp	Ref	Expression
1		ndalle-P7.18.1	. 2 ⊢ (𝛾 → ∀𝑥𝜑)
2		specpsub-P6 721	. 2 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
3	1, 2	syl-P3.24.RC 260	1 ⊢ (𝛾 → [𝑡 / 𝑥]𝜑)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  [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

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155  df-psub-D6.2 716

This theorem is referenced by:  ndalle-P7.18.RC  885  ndalle-P7.18.CL  909  alle-P7  941