Theorem freq12d 42466

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 5650	. . 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 5651	. . 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 278	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   Fr wfr 5632

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 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-v 3473  df-in 3954  df-ss 3964  df-br 5151  df-fr 5635

This theorem is referenced by: (None)