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

Theorem sbequiALT 2598
 Description: Alternate version of sbequi 2092. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dfsb1.xz (𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))
dfsb1.yz (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))
Assertion
Ref Expression
sbequiALT (𝑥 = 𝑦 → (𝜃𝜏))

Proof of Theorem sbequiALT
StepHypRef Expression
1 equtr 2029 . . 3 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
2 dfsb1.xz . . . . 5 (𝜃 ↔ ((𝑧 = 𝑥𝜑) ∧ ∃𝑧(𝑧 = 𝑥𝜑)))
32sbequ2ALT 2582 . . . 4 (𝑧 = 𝑥 → (𝜃𝜑))
4 dfsb1.yz . . . . 5 (𝜏 ↔ ((𝑧 = 𝑦𝜑) ∧ ∃𝑧(𝑧 = 𝑦𝜑)))
54sbequ1ALT 2581 . . . 4 (𝑧 = 𝑦 → (𝜑𝜏))
63, 5syl9 77 . . 3 (𝑧 = 𝑥 → (𝑧 = 𝑦 → (𝜃𝜏)))
71, 6syld 47 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝜃𝜏)))
8 ax13 2395 . . 3 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9 sp 2184 . . . . . 6 (∀𝑧 𝑧 = 𝑥𝑧 = 𝑥)
109con3i 157 . . . . 5 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
112sb4ALT 2590 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → (𝜃 → ∀𝑧(𝑧 = 𝑥𝜑)))
1210, 11syl 17 . . . 4 𝑧 = 𝑥 → (𝜃 → ∀𝑧(𝑧 = 𝑥𝜑)))
13 equeuclr 2031 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
1413imim1d 82 . . . . . 6 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝜑) → (𝑧 = 𝑦𝜑)))
1514al2imi 1817 . . . . 5 (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥𝜑) → ∀𝑧(𝑧 = 𝑦𝜑)))
164sb2ALT 2589 . . . . 5 (∀𝑧(𝑧 = 𝑦𝜑) → 𝜏)
1715, 16syl6 35 . . . 4 (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥𝜑) → 𝜏))
1812, 17syl9 77 . . 3 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑦 → (𝜃𝜏)))
198, 18syld 47 . 2 𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝜃𝜏)))
207, 19pm2.61i 185 1 (𝑥 = 𝑦 → (𝜃𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  sbequALT  2599
 Copyright terms: Public domain W3C validator