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  2852  cgsex4g  2853  vtocl2  2872  vtocl3  2873  dtruarb  4306  opelopabsb  4380  mosubopt  4817  xpmlem  5185  brabvv  6101  ssoprab2i  6144  dmaddpqlem  7697  nqpi  7698  dmaddpq  7699  dmmulpq  7700  enq0sym  7752  enq0ref  7753  enq0tr  7754  nq0nn  7762  prarloc  7823  bj-inex  16726