Theorem 19.41rg 42059

Description: Closed form of right-to-left implication of 19.41 2231, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 42411. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.41rg	⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))

Proof of Theorem 19.41rg

Step	Hyp	Ref	Expression
1		sp 2178	. . . 4 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
2		pm3.21 471	. . . . . . 7 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
3	2	a1i 11	. . . . . 6 ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))))
4	3	al2imi 1819	. . . . 5 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))))
5		exim 1837	. . . . 5 ⊢ (∀𝑥(𝜑 → (𝜑 ∧ 𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
6	4, 5	syl6 35	. . . 4 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))))
7	1, 6	syld 47	. . 3 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))))
8	7	com23 86	. 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑 ∧ 𝜓))))
9	8	impd 410	1 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784

This theorem is referenced by:  ax6e2nd  42067  ax6e2ndVD  42417  ax6e2ndALT  42439