Theorem isoeq145d 42914

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 7321	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))
4		isoeq145.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
5		isoeq4 7324	. . 3 ⊢ (𝐴 = 𝐶 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
7		isoeq145.5	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
8		isoeq5 7325	. . 3 ⊢ (𝐵 = 𝐷 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
10	3, 6, 9	3bitrd 304	1 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   Isom wiso 6544

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 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552

This theorem is referenced by:  resisoeq45d  42915