Theorem 2eximi 1612

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 1611	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦𝜓)
3	2	eximi 1611	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excomim  1674  cgsex2g  2796  cgsex4g  2797  vtocl2  2816  vtocl3  2817  dtruarb  4221  opelopabsb  4291  mosubopt  4725  xpmlem  5087  brabvv  5965  ssoprab2i  6008  dmaddpqlem  7439  nqpi  7440  dmaddpq  7441  dmmulpq  7442  enq0sym  7494  enq0ref  7495  enq0tr  7496  nq0nn  7504  prarloc  7565  bj-inex  15469