Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rr-phpd Structured version   Visualization version   GIF version

Theorem rr-phpd 40840
Description: Equivalent of php 8698 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
rr-phpd.1 (𝜑𝐴 ∈ ω)
rr-phpd.2 (𝜑𝐵𝐴)
rr-phpd.3 (𝜑𝐴𝐵)
Assertion
Ref Expression
rr-phpd (𝜑𝐴 = 𝐵)

Proof of Theorem rr-phpd
StepHypRef Expression
1 rr-phpd.3 . 2 (𝜑𝐴𝐵)
2 rr-phpd.1 . . . 4 (𝜑𝐴 ∈ ω)
3 rr-phpd.2 . . . . . 6 (𝜑𝐵𝐴)
43adantr 484 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
5 simpr 488 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
65neqcomd 2834 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴)
7 dfpss2 4048 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
84, 6, 7sylanbrc 586 . . . 4 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
9 php 8698 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
102, 8, 9syl2an2r 684 . . 3 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵)
1110ex 416 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵))
121, 11mt4d 117 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wss 3919  wpss 3920   class class class wbr 5052  ωcom 7574  cen 8502
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-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-er 8285  df-en 8506  df-dom 8507
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator