Theorem cbvex4v 2409

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker cbvex4vw 2037 if possible. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.)

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 1919	. . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤𝜓))
3	2	cbvex2vv 2408	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑧∃𝑤𝜓)
4		cbvex4v.2	. . . 4 ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))
5	4	cbvex2vv 2408	. . 3 ⊢ (∃𝑧∃𝑤𝜓 ↔ ∃𝑓∃𝑔𝜒)
6	5	2exbii 1843	. 2 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤𝜓 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)
7	3, 6	bitri 274	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-11 2146  ax-12 2166  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)