Theorem 19.41 2202

Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1927 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 1868	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		19.41.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2170	. . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 622	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
5	1, 4	sylib 219	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓))
6		pm3.21 472	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
7	2, 6	eximd 2181	. . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
8	7	impcom 408	. 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
9	5, 8	impbii 210	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1761  Ⅎwnf 1765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-12 2141

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-nf 1766

This theorem is referenced by:  19.42  2203  equsexv  2232  eean  2325  eeeanv  2327  equsexALT  2397  2sb5rf  2453  r19.41  3309  eliunxp  5594  dfopab2  7606  dfoprab3s  7607  xpcomco  8454  mpomptxf  30115  bnj605  31795  bnj607  31804  2sb5nd  40433  2sb5ndVD  40783  2sb5ndALT  40805  eliunxp2  43860