Theorem bj-cbvalim 34472

Description: A lemma used to prove bj-cbval 34476 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbvalim	⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-cbvalim

Step	Hyp	Ref	Expression
1		ax5e 1919	. . 3 ⊢ (∃𝑥𝜓 → 𝜓)
2	1	ax-gen 1802	. 2 ⊢ ∀𝑦(∃𝑥𝜓 → 𝜓)
3		ax-5 1917	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
4		bj-cbvalimt 34466	. . . 4 ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
5	4	com3l 89	. . 3 ⊢ (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦∃𝑥𝜒 → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
6	5	com14 96	. 2 ⊢ (∀𝑦(∃𝑥𝜓 → 𝜓) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))))
7	2, 3, 6	mp2 9	1 ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 210  df-ex 1787

This theorem is referenced by:  bj-cbvalimi  34474