Theorem exi-P7 951

Description: Simplified '∃' Introduction Law. †

For the original form, using explicit substitution, see ndalli-P7.17 842.

Hypothesis

Ref	Expression
exi-P7.1	⊢ (𝛾 → 𝜑)

Assertion

Ref	Expression
exi-P7	⊢ (𝛾 → ∃𝑥𝜑)

Proof of Theorem exi-P7

Step	Hyp	Ref	Expression
1		exi-P7.1	. . 3 ⊢ (𝛾 → 𝜑)
2		psubid-P7 940	. . . 4 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	bisym-P3.33b.RC 299	. . 3 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑)
4	1, 3	subimr2-P4.RC 543	. 2 ⊢ (𝛾 → [𝑥 / 𝑥]𝜑)
5	4	ndexi-P7.19 844	1 ⊢ (𝛾 → ∃𝑥𝜑)

Syntax hints:  term-obj 1   → 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  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:  exi-P7.CL  952  exi-P7r  995  exi-P7r.RC  996  alleexi-P7  1004  qimeqex-P7-L1  1054  qimeqex-P7-L2  1055