Theorem 2exbidv 1879

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 1836	. 2 ⊢ (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
3	2	exbidv 1836	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃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-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1880  4exbidv  1881  cbvex4v  1946  ceqsex3v  2803  ceqsex4v  2804  copsexg  4274  euotd  4284  elopab  4289  elxpi  4676  relop  4813  cbvoprab3  5995  ov6g  6058  th3qlem1  6693  ltresr  7901  fisumcom2  11584  fprodcom2fi  11772