Theorem breqan12d 3953

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
breqan12i.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
breqan12d	⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))

Proof of Theorem breqan12d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		breqan12i.2	. 2 ⊢ (𝜓 → 𝐶 = 𝐷)
3		breq12 3942	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4	1, 2, 3	syl2an 287	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   class class class wbr 3937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by:  breqan12rd  3954  sosng  4620  isoresbr  5718  isoid  5719  isores3  5724  isoini2  5728  ofrfval  5998  oviec  6543  enqbreq2  7189  ltresr2  7672  axpre-ltadd  7718  leltadd  8233  xltneg  9649  lt2sq  10397  le2sq  10398  cnreim  10782  sqrtle  10840  sqrtlt  10841  absext  10867  reefiso  12906  logltb  13003