Theorem cbvex4v 1853

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.)

Hypotheses

Ref	Expression
cbvex4v.1	⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
cbvex4v.2	⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvex4v	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)

Distinct variable groups:   𝑧,𝑤,𝜒   𝑣,𝑢,𝜑   𝑥,𝑦,𝜓   𝑓,𝑔,𝜓   𝑤,𝑓   𝑧,𝑔   𝑤,𝑢,𝑧,𝑣   𝑥,𝑢,𝑤,𝑧   𝑦,𝑣,𝑤,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)   𝜓(𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)

Proof of Theorem cbvex4v

Step	Hyp	Ref	Expression
1		cbvex4v.1	. . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
2	1	2exbidv 1796	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤𝜓))
3	2	cbvex2v 1847	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑧∃𝑤𝜓)
4		cbvex4v.2	. . . 4 ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))
5	4	cbvex2v 1847	. . 3 ⊢ (∃𝑧∃𝑤𝜓 ↔ ∃𝑓∃𝑔𝜒)
6	5	2exbii 1542	. 2 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤𝜓 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)
7	3, 6	bitri 182	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∃wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  enq0sym  6991  addnq0mo  7006  mulnq0mo  7007  addsrmo  7289  mulsrmo  7290