Theorem 2eximi 1589

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 1588	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦𝜓)
3	2	eximi 1588	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  excomim  1651  cgsex2g  2762  cgsex4g  2763  vtocl2  2781  vtocl3  2782  dtruarb  4170  opelopabsb  4238  mosubopt  4669  xpmlem  5024  brabvv  5888  ssoprab2i  5931  dmaddpqlem  7318  nqpi  7319  dmaddpq  7320  dmmulpq  7321  enq0sym  7373  enq0ref  7374  enq0tr  7375  nq0nn  7383  prarloc  7444  bj-inex  13789