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

Theorem cbvexdvaOLD 2272
 Description: Obsolete proof of cbvexdva 2271 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvaldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvexdvaOLD (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvexdvaOLD
StepHypRef Expression
1 nfv 1830 . 2 𝑦𝜑
2 nfvd 1831 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 cbvaldva.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 449 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4cbvexd 2266 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator