Theorem isoeq145d 44036

Description: Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)

Hypotheses

Ref	Expression
isoeq145.1	⊢ (𝜑 → 𝐹 = 𝐺)
isoeq145.4	⊢ (𝜑 → 𝐴 = 𝐶)
isoeq145.5	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
isoeq145d	⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Proof of Theorem isoeq145d

Step	Hyp	Ref	Expression
1		isoeq145.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2		isoeq1 7316	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
4		isoeq145.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
5		isoeq4 7319	. . 3 ⊢ (𝐴 = 𝐶 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
7		isoeq145.5	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
8		isoeq5 7320	. . 3 ⊢ (𝐵 = 𝐷 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
10	3, 6, 9	3bitrd 308	1 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   Isom wiso 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  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:  resisoeq45d  44037