Theorem lemma-L6.07a-L2 771

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

Assertion

Ref	Expression
lemma-L6.07a-L2	⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.07a-L2

Step	Hyp	Ref	Expression
1		rcp-NDASM1of2 193	. . . . 5 ⊢ ((𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) → 𝑥 = 𝑡)
2		lemma-L6.01a 724	. . . . . . 7 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
3	2	ndbier-P3.15 180	. . . . . 6 ⊢ (𝑥 = 𝑡 → (∀𝑥(𝑥 = 𝑡 → 𝜑) → 𝜑))
4	3	import-P3.34a.RC 306	. . . . 5 ⊢ ((𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) → 𝜑)
5	1, 4	ndandi-P3.7 172	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (𝑥 = 𝑡 ∧ 𝜑))
6	5	alloverimex-P5.RC.GEN 603	. . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))
7		nfrall1-P6 741	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)
8	7	qcexandl-P6 762	. . . 4 ⊢ ((∃𝑥 𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∃𝑥(𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
9	8	bisym-P3.33b.RC 299	. . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑥 𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
10	6, 9	subiml2-P4.RC 541	. 2 ⊢ ((∃𝑥 𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))
11		axL6ex-P5 625	. . 3 ⊢ ∃𝑥 𝑥 = 𝑡
12	11	idandthml-P4.23a 446	. 2 ⊢ ((∃𝑥 𝑥 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
13	10, 12	subiml2-P4.RC 541	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