Theorem 2eximi 1492

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 1491	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦𝜓)
3	2	eximi 1491	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  excomim  1553  cgsex2g  2590  cgsex4g  2591  vtocl2  2609  vtocl3  2610  dtruarb  3942  opelopabsb  3997  mosubopt  4405  xpmlem  4744  brabvv  5551  ssoprab2i  5593  dmaddpqlem  6475  nqpi  6476  dmaddpq  6477  dmmulpq  6478  enq0sym  6530  enq0ref  6531  enq0tr  6532  nq0nn  6540  prarloc  6601  bj-inex  10027