Theorem alle-P7 941

Description: Simplified '∀' Elimination Law. †

This is also known as the Specialization Law. For the original form, using explicit substitution, see ndalle-P7.18 843.

Hypothesis

Ref	Expression
alle-P7.1	⊢ (𝛾 → ∀𝑥𝜑)

Assertion

Ref	Expression
alle-P7	⊢ (𝛾 → 𝜑)

Proof of Theorem alle-P7

Step	Hyp	Ref	Expression
1		alle-P7.1	. . 3 ⊢ (𝛾 → ∀𝑥𝜑)
2	1	ndalle-P7.18 843	. 2 ⊢ (𝛾 → [𝑥 / 𝑥]𝜑)
3		psubid-P7 940	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	subimr2-P4.RC 543	1 ⊢ (𝛾 → 𝜑)

Syntax hints:  term-obj 1  ∀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  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:  alle-P7.CL  942  alle-P7r  992  alle-P7r.RC  993  alleexi-P7  1004  exnegallint-P7  1047  qimeqex-P7-L2  1055