Theorem lemma-L6.07a-L1 770

Description: Lemma for lemma-L6.06a 766.

Assertion

Ref	Expression
lemma-L6.07a-L1	⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.07a-L1

Step	Hyp	Ref	Expression
1		lemma-L6.01a 724	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2	1	ndbief-P3.14 179	. . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
3	2	import-P3.34a.RC 306	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
4	3	alloverimex-P5.RC.GEN 603	. 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))
5		nfrall1-P6 741	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)
6	5	qremex-P6 723	. 2 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
7	4, 6	subimr2-P4.RC 543	1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132  ∃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-L10 27  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

This theorem is referenced by:  lemma-L6.07a  772