Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.8aOLD Structured version   Visualization version   GIF version

Theorem 19.8aOLD 2040
 Description: Obsolete proof of 19.8a 2039. Obsolete as of 21-Dec-2020. Can be deleted as soon as the question of why "MM-PA> min exlimiiv" does not give 19.8a 2039 is answered. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2041. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.8aOLD (𝜑 → ∃𝑥𝜑)

Proof of Theorem 19.8aOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1877 . 2 𝑦 𝑦 = 𝑥
2 ax12v 2035 . . . . 5 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 ax6ev 1877 . . . . 5 𝑥 𝑥 = 𝑦
4 exim 1751 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
52, 3, 4syl6mpi 65 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∃𝑥𝜑))
65equcoms 1934 . . 3 (𝑦 = 𝑥 → (𝜑 → ∃𝑥𝜑))
76exlimiv 1845 . 2 (∃𝑦 𝑦 = 𝑥 → (𝜑 → ∃𝑥𝜑))
81, 7ax-mp 5 1 (𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃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-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator