Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-equsal1i Structured version   Visualization version   GIF version

Theorem wl-equsal1i 33459
Description: The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)
Hypotheses
Ref Expression
wl-equsal1i.1 (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
wl-equsal1i.2 (𝑥 = 𝑦𝜑)
Assertion
Ref Expression
wl-equsal1i 𝜑

Proof of Theorem wl-equsal1i
StepHypRef Expression
1 wl-equsal1i.1 . 2 (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
2 wl-equsal1i.2 . . 3 (𝑥 = 𝑦𝜑)
32gen2 1763 . 2 𝑥𝑦(𝑥 = 𝑦𝜑)
4 sp 2091 . . . . 5 (∀𝑦𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
54alcoms 2075 . . . 4 (∀𝑥𝑦(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
6 wl-equsal1t 33457 . . . 4 (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
75, 6syl5ib 234 . . 3 (Ⅎ𝑥𝜑 → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
8 wl-equsalcom 33458 . . . . 5 (∀𝑦(𝑦 = 𝑥𝜑) ↔ ∀𝑦(𝑥 = 𝑦𝜑))
9 wl-equsal1t 33457 . . . . . 6 (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥𝜑) ↔ 𝜑))
109biimpd 219 . . . . 5 (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥𝜑) → 𝜑))
118, 10syl5bir 233 . . . 4 (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦𝜑) → 𝜑))
1211spsd 2095 . . 3 (Ⅎ𝑦𝜑 → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
137, 12jaoi 393 . 2 ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥𝑦(𝑥 = 𝑦𝜑) → 𝜑))
141, 3, 13mp2 9 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator