Theorem 4exbidv 1918

Description: Formula-building rule for 4 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 1916	. 2 ⊢ (𝜑 → (∃𝑧∃𝑤𝜓 ↔ ∃𝑧∃𝑤𝜒))
3	2	2exbidv 1916	1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1540

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ceqsex8v  2849  copsex4g  4339  opbrop  4805  ovi3  6159  brecop  6794  th3q  6809  dfplpq2  7574  dfmpq2  7575  enq0sym  7652  enq0ref  7653  enq0tr  7654  enq0breq  7656  addnq0mo  7667  mulnq0mo  7668  addnnnq0  7669  mulnnnq0  7670  addsrmo  7963  mulsrmo  7964  addsrpr  7965  mulsrpr  7966