Theorem isoeq145d 44145

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 7315	. . 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 7318	. . 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 7319	. . 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 1570   Isom wiso 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545

This theorem is referenced by:  resisoeq45d  44146