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Theorem isoeq145d 43410
Description: Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
Hypotheses
Ref Expression
isoeq145.1 (𝜑𝐹 = 𝐺)
isoeq145.4 (𝜑𝐴 = 𝐶)
isoeq145.5 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
isoeq145d (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Proof of Theorem isoeq145d
StepHypRef Expression
1 isoeq145.1 . . 3 (𝜑𝐹 = 𝐺)
2 isoeq1 7335 . . 3 (𝐹 = 𝐺 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
31, 2syl 17 . 2 (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
4 isoeq145.4 . . 3 (𝜑𝐴 = 𝐶)
5 isoeq4 7338 . . 3 (𝐴 = 𝐶 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
64, 5syl 17 . 2 (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
7 isoeq145.5 . . 3 (𝜑𝐵 = 𝐷)
8 isoeq5 7339 . . 3 (𝐵 = 𝐷 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
97, 8syl 17 . 2 (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
103, 6, 93bitrd 305 1 (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540   Isom wiso 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-isom 6568
This theorem is referenced by:  resisoeq45d  43411
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