Theorem 4exbidv 1841

Description: Formula-building rule for four existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)

Hypothesis

Ref	Expression
4exbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
4exbidv	⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧   𝜑,𝑤

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

Proof of Theorem 4exbidv

Step	Hyp	Ref	Expression
1		4exbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	2exbidv 1839	. 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒))
3	2	2exbidv 1839	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))

Syntax hints:   → wi 4   ↔ wb 195  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  ceqsex8v  3222  copsex4g  4885  opbrop  5121  ov3  6695  brecop  7727  addsrmo  9773  mulsrmo  9774  addsrpr  9775  mulsrpr  9776  dihopelvalcpre  35555  xihopellsmN  35561  dihopellsm  35562