Theorem bj-cbv3tb 37137

Description: Closed form of cbv3 2401. (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 2212	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
2	1	biimpd 230	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
3	2	alimi 1814	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦(∃𝑥𝜓 → 𝜓))
4		nf5r 2202	. . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑))
5	4	alimi 1814	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑))
6		bj-cbv3ta 37136	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓)))
7	3, 5, 6	syl2ani 609	1 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓)))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1541  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-11 2164  ax-12 2185  ax-13 2376

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)