Theorem 19.41 2277

Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1976 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 1913	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		19.41.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2247	. . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 634	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
5	1, 4	sylib 221	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓))
6		pm3.21 476	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
7	2, 6	eximd 2258	. . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
8	7	impcom 412	. 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
9	5, 8	impbii 212	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1806  Ⅎwnf 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811

This theorem is referenced by:  19.42  2278  eean  2386  eeeanv  2388  equsexALT  2457  2sb5rf  2510  r19.41  3275  eliunxp  5821  dfopab2  8045  dfoprab3s  8046  xpcomco  9051  mpomptxf  32960  bnj605  35236  bnj607  35245  2sb5nd  45156  2sb5ndVD  45505  2sb5ndALT  45527  eliunxp2  48994