Theorem heeq2 44318

Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
heeq2	⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵))

Proof of Theorem heeq2

Step	Hyp	Ref	Expression
1		eqid 2761	. 2 ⊢ 𝑅 = 𝑅
2		heeq12 44316	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵))
3	1, 2	mpan 700	1 ⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   hereditary whe 44312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-he 44313

This theorem is referenced by: (None)