Theorem freq12d 37609

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 5084	. . 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 5085	. . 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 268	1 ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-in 3581  df-ss 3588  df-br 4654  df-fr 5073

This theorem is referenced by: (None)