Theorem exi-P6 718

Description: Existential Quantifier Introduction Law.

See exiw-P5 662 for a version that requires only FOL axioms.

Assertion

Ref	Expression
exi-P6	⊢ (𝜑 → ∃𝑥𝜑)

Proof of Theorem exi-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-L12 29	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		alloverimex-P5.CL 604	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
3	2	rcp-NDIMP0addall 207	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)))
4	1, 3	syl-P3.24 259	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)))
5		axL6ex-P5 625	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
6	5	rcp-NDIMP0addall 207	. . 3 ⊢ (𝑥 = 𝑦 → ∃𝑥 𝑥 = 𝑦)
7	4, 6	mae-P3.23 257	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥𝜑))
8		axL6ex-P5 625	. . 3 ⊢ ∃𝑦 𝑦 = 𝑥
9		eqsym-P5.CL.SYM 629	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
10	9	subexinf-P5 608	. . 3 ⊢ (∃𝑦 𝑥 = 𝑦 ↔ ∃𝑦 𝑦 = 𝑥)
11	8, 10	bimpr-P4.RC 534	. 2 ⊢ ∃𝑦 𝑥 = 𝑦
12	7, 11	exiav-P5.SH 616	1 ⊢ (𝜑 → ∃𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10  ∃wff-exists 595

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-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

This theorem is referenced by:  spec-P6  719  qremall-P6  722  qremex-P6  723  nfrgen-P6  733  spliteq-P6-L1  775  splitelof-P6-L1  777  nfrgencl-L6  811  qremexd-P6  823