Theorem exipsub-P6 720

Description: Existential Quantifier Introduction Law (proper substitution).

This is the form most often seen in logic text books.

Assertion

Ref	Expression
exipsub-P6	⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem exipsub-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-psub-D6.2 716	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	rcp-NDBIEF0 240	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		alloverimex-P5.CL 604	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦 𝑦 = 𝑡 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		axL6ex-P5 625	. . 3 ⊢ ∃𝑦 𝑦 = 𝑡
5	3, 4	mae-P3.23.RC 258	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
6		alloverimex-P5.CL 604	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
7		axL6ex-P5 625	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
8	6, 7	mae-P3.23.RC 258	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
9	8	exiav-P5 615	. 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
10	2, 5, 9	dsyl-P3.25.RC 262	1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → 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  844