Theorem 4exbidv 1959

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 1957	. 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒))
3	2	2exbidv 1957	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  ceqsex8v  3506  copsex4g  5467  opbrop  5749  ov3  7583  brecop  8831  addsrmo  11158  mulsrmo  11159  addsrpr  11160  mulsrpr  11161  dihopelvalcpre  42305  xihopellsmN  42311  dihopellsm  42312