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  2837  cgsex4g  2838  vtocl2  2857  vtocl3  2858  dtruarb  4279  opelopabsb  4352  mosubopt  4789  xpmlem  5155  brabvv  6062  ssoprab2i  6105  dmaddpqlem  7590  nqpi  7591  dmaddpq  7592  dmmulpq  7593  enq0sym  7645  enq0ref  7646  enq0tr  7647  nq0nn  7655  prarloc  7716  bj-inex  16452