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Theorem rr-phpd 44112
Description: Equivalent of php 9269 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 480 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
5 simpr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
65neqcomd 2744 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴)
7 dfpss2 4105 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
84, 6, 7sylanbrc 582 . . . 4 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
9 php 9269 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
102, 8, 9syl2an2r 684 . . 3 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵)
1110ex 412 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵))
121, 11mt4d 117 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2103  wss 3970  wpss 3971   class class class wbr 5169  ωcom 7899  cen 8996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-om 7900  df-1o 8518  df-en 9000  df-dom 9001  df-fin 9003
This theorem is referenced by: (None)
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