Theorem ndexi-P7.19 844

Description: Natural Deduction: '∃' Introduction Rule.

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

'⊢ (𝛾 → 𝜑(𝑡))   ⇒   ⊢ (𝛾 → ∃𝑥𝜑(𝑥))'.

Within a proof, this rule is used to replace a specific constant or functional term with an abstract variable. The use of this rule is indicated by the phrase "therefore there exists".

Hypothesis

Ref	Expression
ndexi-P7.19.1	⊢ (𝛾 → [𝑡 / 𝑥]𝜑)

Assertion

Ref	Expression
ndexi-P7.19	⊢ (𝛾 → ∃𝑥𝜑)

Proof of Theorem ndexi-P7.19

Step	Hyp	Ref	Expression
1		ndexi-P7.19.1	. 2 ⊢ (𝛾 → [𝑡 / 𝑥]𝜑)
2		exipsub-P6 720	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
3	1, 2	syl-P3.24.RC 260	1 ⊢ (𝛾 → ∃𝑥𝜑)

Syntax hints:   → wff-imp 10  ∃wff-exists 595  [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

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-psub-D6.2 716

This theorem is referenced by:  ndexi-P7.19.RC  886  ndexi-P7.19.CL  910  exi-P7  951