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Theorem resisoeq45d 42752
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 5975 . 2 (𝜑 → (𝐹𝐴) = (𝐹𝐶))
3 resisoeq45.5 . 2 (𝜑𝐵 = 𝐷)
42, 1, 3isoeq145d 42751 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 205   = wceq 1533  cres 5671   Isom wiso 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546
This theorem is referenced by:  negslem1  42753
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