Theorem cbvall-P7-L1 1060

Description: Lemma for cbvall-P7 1061. †

Hypotheses

Ref	Expression
cbvall-P7-L1.1	⊢ Ⅎ𝑥𝜓
cbvall-P7-L1.2	⊢ Ⅎ𝑦𝜑
cbvall-P7-L1.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvall-P7-L1	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Proof of Theorem cbvall-P7-L1

Step	Hyp	Ref	Expression
1		cbvall-P7-L1.2	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	ndnfrall2-P7.9.RC 880	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
3		ndalle-P7.18.CL 909	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
4		cbvall-P7-L1.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
5		cbvall-P7-L1.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	4, 5	ndpsub1-P7.13 838	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7	6	rcp-NDBIEF0 240	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)
8	3, 7	syl-P3.24.RC 260	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
9	2, 8	alli-P7 947	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  Ⅎwff-nfree 681  [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:  cbvall-P7  1061