Theorem 2eximi 1647

Description: Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)

Hypothesis

Ref	Expression
eximi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
2eximi	⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Proof of Theorem 2eximi

Step	Hyp	Ref	Expression
1		eximi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	eximi 1646	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦𝜓)
3	2	eximi 1646	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excomim  1709  cgsex2g  2836  cgsex4g  2837  vtocl2  2856  vtocl3  2857  dtruarb  4276  opelopabsb  4349  mosubopt  4786  xpmlem  5152  brabvv  6059  ssoprab2i  6102  dmaddpqlem  7580  nqpi  7581  dmaddpq  7582  dmmulpq  7583  enq0sym  7635  enq0ref  7636  enq0tr  7637  nq0nn  7645  prarloc  7706  bj-inex  16379