Theorem freq12d 40508

Description: Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
weeq12d.l	⊢ (𝜑 → 𝑅 = 𝑆)
weeq12d.r	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
freq12d	⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Proof of Theorem freq12d

Step	Hyp	Ref	Expression
1		weeq12d.l	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
2		freq1 5506	. . 3 ⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
4		weeq12d.r	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
5		freq2 5507	. . 3 ⊢ (𝐴 = 𝐵 → (𝑆 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (𝑆 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
7	3, 6	bitrd 282	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543   Fr wfr 5491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-v 3400  df-in 3860  df-ss 3870  df-br 5040  df-fr 5494

This theorem is referenced by: (None)