Theorem 2exbidv 1892

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 1849	. 2 ⊢ (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒))
3	2	exbidv 1849	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1516

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1893  4exbidv  1894  cbvex4v  1959  ceqsex3v  2817  ceqsex4v  2818  copsexg  4295  euotd  4306  elopab  4311  elxpi  4698  relop  4835  cbvoprab3  6033  ov6g  6096  th3qlem1  6736  ltresr  7967  fisumcom2  11819  fprodcom2fi  12007