MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbval2 Structured version   Visualization version   GIF version

Theorem cbval2 2278
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
Hypotheses
Ref Expression
cbval2.1 𝑧𝜑
cbval2.2 𝑤𝜑
cbval2.3 𝑥𝜓
cbval2.4 𝑦𝜓
cbval2.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbval2 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧   𝑥,𝑤   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbval2
StepHypRef Expression
1 cbval2.1 . . 3 𝑧𝜑
21nfal 2150 . 2 𝑧𝑦𝜑
3 cbval2.3 . . 3 𝑥𝜓
43nfal 2150 . 2 𝑥𝑤𝜓
5 nfv 1840 . . . . . 6 𝑤 𝑥 = 𝑧
6 cbval2.2 . . . . . 6 𝑤𝜑
75, 6nfim 1822 . . . . 5 𝑤(𝑥 = 𝑧𝜑)
8 nfv 1840 . . . . . 6 𝑦 𝑥 = 𝑧
9 cbval2.4 . . . . . 6 𝑦𝜓
108, 9nfim 1822 . . . . 5 𝑦(𝑥 = 𝑧𝜓)
11 cbval2.5 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1211expcom 451 . . . . . 6 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
1312pm5.74d 262 . . . . 5 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
147, 10, 13cbval 2270 . . . 4 (∀𝑦(𝑥 = 𝑧𝜑) ↔ ∀𝑤(𝑥 = 𝑧𝜓))
15 19.21v 1865 . . . 4 (∀𝑦(𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑))
16 19.21v 1865 . . . 4 (∀𝑤(𝑥 = 𝑧𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
1714, 15, 163bitr3i 290 . . 3 ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
1817pm5.74ri 261 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
192, 4, 18cbval 2270 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  cbvex2  2279  cbval2vOLD  2285
  Copyright terms: Public domain W3C validator