Theorem 2eximi 1650

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 1649	. 2 ⊢ (∃𝑦𝜑 → ∃𝑦𝜓)
3	2	eximi 1649	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∃wex 1541

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  excomim  1711  cgsex2g  2849  cgsex4g  2850  vtocl2  2869  vtocl3  2870  dtruarb  4303  opelopabsb  4377  mosubopt  4814  xpmlem  5182  brabvv  6098  ssoprab2i  6141  dmaddpqlem  7688  nqpi  7689  dmaddpq  7690  dmmulpq  7691  enq0sym  7743  enq0ref  7744  enq0tr  7745  nq0nn  7753  prarloc  7814  bj-inex  16664