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

Theorem spfwOLD 1963
Description: Obsolete proof of spfw 1962 as of 10-Oct-2021. (Contributed by NM, 19-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
spfw.1 𝜓 → ∀𝑥 ¬ 𝜓)
spfw.2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
spfw.3 𝜑 → ∀𝑦 ¬ 𝜑)
spfw.4 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spfwOLD (∀𝑥𝜑𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spfwOLD
StepHypRef Expression
1 spfw.2 . . 3 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
2 alim 1735 . . 3 (∀𝑦(∀𝑥𝜑𝜓) → (∀𝑦𝑥𝜑 → ∀𝑦𝜓))
3 spfw.3 . . . 4 𝜑 → ∀𝑦 ¬ 𝜑)
4 spfw.4 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
54biimprd 238 . . . . 5 (𝑥 = 𝑦 → (𝜓𝜑))
65equcoms 1944 . . . 4 (𝑦 = 𝑥 → (𝜓𝜑))
73, 6spimw 1923 . . 3 (∀𝑦𝜓𝜑)
81, 2, 7syl56 36 . 2 (∀𝑦(∀𝑥𝜑𝜓) → (∀𝑥𝜑𝜑))
9 spfw.1 . . 3 𝜓 → ∀𝑥 ¬ 𝜓)
104biimpd 219 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
119, 10spimw 1923 . 2 (∀𝑥𝜑𝜓)
128, 11mpg 1721 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478
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
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator