Theorem lemma-L6.01a 724

Description: This lemma is the result of solvesub-P6b 707 with no hypothesis required.

'𝑥' cannot occur in '𝑡'.

Assertion

Ref	Expression
lemma-L6.01a	⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.01a

Step	Hyp	Ref	Expression
1		ax-L12 29	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2		spec-P6 719	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → (𝑥 = 𝑡 → 𝜑))
3	2	imcomm-P3.27.RC 266	. 2 ⊢ (𝑥 = 𝑡 → (∀𝑥(𝑥 = 𝑡 → 𝜑) → 𝜑))
4	1, 3	ndbii-P3.13 178	1 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104

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:  psubtoisubv-P6  725  lemma-L6.05a  764  lemma-L6.07a-L1  770  lemma-L6.07a-L2  771