Theorem bj-indeq 14766

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 2241	. . 3 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
2		eleq2 2241	. . . 4 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵))
3	2	raleqbi1dv 2681	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵))
4	1, 3	anbi12d 473	. 2 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)))
5		df-bj-ind 14764	. 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
6		df-bj-ind 14764	. 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 1353   ∈ wcel 2148  ∀wral 2455  ∅c0 3424  suc csuc 4367  Ind wind 14763

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-bj-ind 14764

This theorem is referenced by:  bj-omind  14771  bj-omssind  14772  bj-ssom  14773  bj-om  14774  bj-2inf  14775