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Theorem cbval2 2433
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker cbval2v 2364 if possible. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
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 2343 . 2 𝑧𝑦𝜑
3 cbval2.3 . . 3 𝑥𝜓
43nfal 2343 . 2 𝑥𝑤𝜓
5 nfv 1916 . . 3 𝑦 𝑥 = 𝑧
6 nfv 1916 . . 3 𝑤 𝑥 = 𝑧
7 cbval2.2 . . . 4 𝑤𝜑
87a1i 11 . . 3 (𝑥 = 𝑧 → Ⅎ𝑤𝜑)
9 cbval2.4 . . . 4 𝑦𝜓
109a1i 11 . . 3 (𝑥 = 𝑧 → Ⅎ𝑦𝜓)
11 cbval2.5 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1211ex 416 . . 3 (𝑥 = 𝑧 → (𝑦 = 𝑤 → (𝜑𝜓)))
135, 6, 8, 10, 12cbv2 2424 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
142, 4, 13cbval 2417 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbvex2  2435
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