Theorem bj-indeq 16869

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 2302	. . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
2		eleq2 2302	. . . 4 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵))
3	2	raleqbi1dv 2761	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵))
4	1, 3	anbi12d 477	. 2 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)))
5		df-bj-ind 16867	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
6		df-bj-ind 16867	. 2 ⊢ (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵))
7	4, 5, 6	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∅c0 3520  suc csuc 4505  Ind wind 16866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-bj-ind 16867

This theorem is referenced by:  bj-omind  16874  bj-omssind  16875  bj-ssom  16876  bj-om  16877  bj-2inf  16878