Theorem lemma-L6.04a 749

Description: Version of lemma-L5.02a 653 with non-freeness condition.

'𝑥 cannot occur in '𝑡'.

Hypotheses

Ref	Expression
lemma-L6.04a.1	⊢ Ⅎ𝑥𝜓
lemma-L6.04a.2	⊢ (𝑥 = 𝑡 → (𝜑 → 𝜓))

Assertion

Ref	Expression
lemma-L6.04a	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.04a

Step	Hyp	Ref	Expression
1		lemma-L6.04a.2	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 → 𝜓))
2	1	imcomm-P3.27.RC 266	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑡 → 𝜓))
3	2	dalloverimex-P5.RC.GEN 607	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝑡 → ∃𝑥𝜓))
4		axL6ex-P5 625	. . 3 ⊢ ∃𝑥 𝑥 = 𝑡
5	3, 4	mae-P3.23.RC 258	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
6		lemma-L6.04a.1	. . 3 ⊢ Ⅎ𝑥𝜓
7	6	nfrexgen-P6 735	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
8	5, 7	syl-P3.24.RC 260	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10  ∃wff-exists 595  Ⅎwff-nfree 681

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  df-nfree-D6.1 682

This theorem is referenced by:  cbvall-P6-L1  750