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Theorem rr-phpd 42575
Description: Equivalent of php 9160 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 482 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
5 simpr 486 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
65neqcomd 2743 . . . . 5 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴)
7 dfpss2 4049 . . . . 5 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐵 = 𝐴))
84, 6, 7sylanbrc 584 . . . 4 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵𝐴)
9 php 9160 . . . 4 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
102, 8, 9syl2an2r 684 . . 3 ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴𝐵)
1110ex 414 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝐴𝐵))
121, 11mt4d 117 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3914  wpss 3915   class class class wbr 5109  ωcom 7806  cen 8886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-en 8890  df-dom 8891  df-fin 8893
This theorem is referenced by: (None)
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