Theorem 19.41 2238

Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1950 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)

Hypothesis

Ref	Expression
19.41.1	⊢ Ⅎ𝑥𝜓

Assertion

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

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		19.40 1887	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		19.41.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2208	. . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 623	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
5	1, 4	sylib 218	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓))
6		pm3.21 471	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
7	2, 6	eximd 2219	. . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
8	7	impcom 407	. 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
9	5, 8	impbii 209	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  19.42  2239  eean  2348  eeeanv  2350  equsexALT  2419  2sb5rf  2472  r19.41  3236  eliunxp  5772  dfopab2  7979  dfoprab3s  7980  xpcomco  8975  mpomptxf  32651  bnj605  34911  bnj607  34920  2sb5nd  44593  2sb5ndVD  44942  2sb5ndALT  44964  eliunxp2  48365