Theorem 2exbidv 1856

Description: Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)

Hypothesis

Ref	Expression
2albidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
2exbidv	⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem 2exbidv

Step	Hyp	Ref	Expression
1		2albidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	exbidv 1813	. 2 ⊢ (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
3	2	exbidv 1813	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∃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-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  3exbidv  1857  4exbidv  1858  cbvex4v  1918  ceqsex3v  2768  ceqsex4v  2769  copsexg  4222  euotd  4232  elopab  4236  elxpi  4620  relop  4754  cbvoprab3  5918  ov6g  5979  th3qlem1  6603  ltresr  7780  fisumcom2  11379  fprodcom2fi  11567