Theorem specpsub-P6 721

Description: Law of Specialization. (proper substitution).

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

Assertion

Ref	Expression
specpsub-P6	⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Proof of Theorem specpsub-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axL1-P3.21.CL 253	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑))
2	1	alloverim-P5.RC.GEN 592	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	axL1-P3.21 252	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	allicv-P5 614	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5		df-psub-D6.2 716	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	bisym-P3.33b.RC 299	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ [𝑡 / 𝑥]𝜑)
7	4, 6	subimr2-P4.RC 543	1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀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  843