Theorem 4exbidv 1927

Description: Formula-building rule for four existential quantifiers (deduction form). (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 1925	. 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒))
3	2	2exbidv 1925	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  ceqsex8v  3494  copsex4g  5433  opbrop  5712  ov3  7509  brecop  8734  addsrmo  10964  mulsrmo  10965  addsrpr  10966  mulsrpr  10967  dihopelvalcpre  41357  xihopellsmN  41363  dihopellsm  41364