Theorem 19.41 2236

Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1949 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 1886	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2		19.41.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2206	. . . 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 2217	. . 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 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  19.42  2237  eean  2346  eeeanv  2348  equsexALT  2417  2sb5rf  2470  r19.41  3241  eliunxp  5801  dfopab2  8031  dfoprab3s  8032  xpcomco  9031  mpomptxf  32601  bnj605  34897  bnj607  34906  2sb5nd  44550  2sb5ndVD  44899  2sb5ndALT  44921  eliunxp2  48322