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  4275  opelopabsb  4348  mosubopt  4784  xpmlem  5149  brabvv  6056  ssoprab2i  6099  dmaddpqlem  7575  nqpi  7576  dmaddpq  7577  dmmulpq  7578  enq0sym  7630  enq0ref  7631  enq0tr  7632  nq0nn  7640  prarloc  7701  bj-inex  16353