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Theorem cbv2 2416
 Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2383. See cbv2w 2350 with disjoint variable conditions, not depending on ax-13 2383. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style, avoid ax-10 2138. (Revised by Wolf Lammen, 10-Sep-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbv2.1 𝑥𝜑
cbv2.2 𝑦𝜑
cbv2.3 (𝜑 → Ⅎ𝑦𝜓)
cbv2.4 (𝜑 → Ⅎ𝑥𝜒)
cbv2.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.1 . . 3 𝑥𝜑
2 cbv2.2 . . 3 𝑦𝜑
3 cbv2.3 . . 3 (𝜑 → Ⅎ𝑦𝜓)
4 cbv2.4 . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 cbv2.5 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
6 biimp 217 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
75, 6syl6 35 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
81, 2, 3, 4, 7cbv1 2415 . 2 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
9 equcomi 2017 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
10 biimpr 222 . . . 4 ((𝜓𝜒) → (𝜒𝜓))
119, 5, 10syl56 36 . . 3 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
122, 1, 4, 3, 11cbv1 2415 . 2 (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
138, 12impbid 214 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1528  Ⅎwnf 1777 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 1904  ax-6 1963  ax-7 2008  ax-11 2153  ax-12 2169  ax-13 2383 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778 This theorem is referenced by:  cbvald  2421  cbval2  2425  sb9  2555  wl-cbvalnaed  34764  wl-sb8t  34780
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