Theorem bj-indeq 13811

Description: Equality property for Ind. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indeq	⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Proof of Theorem bj-indeq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2230	. . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
2		eleq2 2230	. . . 4 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵))
3	2	raleqbi1dv 2669	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵))
4	1, 3	anbi12d 465	. 2 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)))
5		df-bj-ind 13809	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
6		df-bj-ind 13809	. 2 ⊢ (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵))
7	4, 5, 6	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∅c0 3409  suc csuc 4343  Ind wind 13808

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-bj-ind 13809

This theorem is referenced by:  bj-omind  13816  bj-omssind  13817  bj-ssom  13818  bj-om  13819  bj-2inf  13820