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Theorem resisoeq45d 41766
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 5942 . 2 (𝜑 → (𝐹𝐴) = (𝐹𝐶))
3 resisoeq45.5 . 2 (𝜑𝐵 = 𝐷)
42, 1, 3isoeq145d 41765 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 1542  cres 5640   Isom wiso 6502
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-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510
This theorem is referenced by:  negslem1  41767
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