Theorem 19.41h 1663

Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1664 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.41h.1	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
19.41h	⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Proof of Theorem 19.41h

Step	Hyp	Ref	Expression
1		19.40 1610	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		19.41h.1	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
3		id 19	. . . . 5 ⊢ (𝜓 → 𝜓)
4	2, 3	exlimih 1572	. . . 4 ⊢ (∃𝑥𝜓 → 𝜓)
5	4	anim2i 339	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∃𝑥𝜑 ∧ 𝜓))
6	1, 5	syl 14	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓))
7		pm3.21 262	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
8	2, 7	eximdh 1590	. . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
9	8	impcom 124	. 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
10	6, 9	impbii 125	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.42h  1665  sbh  1749  sbidm  1823  19.41v  1874  2exeu  2091