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Theorem resisoeq45d 43368
Description: Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
Hypotheses
Ref Expression
resisoeq45.4 (𝜑𝐴 = 𝐶)
resisoeq45.5 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
resisoeq45d (𝜑 → ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Proof of Theorem resisoeq45d
StepHypRef Expression
1 resisoeq45.4 . . 3 (𝜑𝐴 = 𝐶)
21reseq2d 5994 . 2 (𝜑 → (𝐹𝐴) = (𝐹𝐶))
3 resisoeq45.5 . 2 (𝜑𝐵 = 𝐷)
42, 1, 3isoeq145d 43367 1 (𝜑 → ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1535  cres 5685   Isom wiso 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567
This theorem is referenced by:  negslem1  43369
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