Theorem cbvallv-P5-L1 658

Description: Lemma for cbvallv-P5 659.

Hypothesis

Ref	Expression
cbvallv-P5-L1.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbvallv-P5-L1	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

Proof of Theorem cbvallv-P5-L1

Step	Hyp	Ref	Expression
1		ax-L5 17	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		cbvallv-P5-L1.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	lemma-L5.02a 653	. . 3 ⊢ (∀𝑥𝜑 → 𝜓)
4	3	alloverim-P5.RC.GEN 592	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑦𝜓)
5	1, 4	syl-P3.24.RC 260	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

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

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

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:  cbvallv-P5  659