Theorem bj-cbv3tb 37741

Description: Closed form of cbv3 2426. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-cbv3tb	⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Proof of Theorem bj-cbv3tb

Step	Hyp	Ref	Expression
1		19.9t 2240	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
2	1	biimpd 232	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	2	alimi 1843	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦(∃𝑥𝜓 → 𝜓))
4		nf5r 2230	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
5	4	alimi 1843	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
6		bj-cbv3ta 37740	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
7	3, 5, 6	syl2ani 619	1 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-11 2193  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816

This theorem is used by: (None)