Theorem seeq1 4357

Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
seeq1	⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))

Proof of Theorem seeq1

Step	Hyp	Ref	Expression
1		eqimss2 3225	. . 3 ⊢ (𝑅 = 𝑆 → 𝑆 ⊆ 𝑅)
2		sess1 4355	. . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴))
3	1, 2	syl 14	. 2 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴))
4		eqimss 3224	. . 3 ⊢ (𝑅 = 𝑆 → 𝑅 ⊆ 𝑆)
5		sess1 4355	. . 3 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))
6	4, 5	syl 14	. 2 ⊢ (𝑅 = 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))
7	3, 6	impbid 129	1 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ⊆ wss 3144   Se wse 4347

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-br 4019  df-se 4351

This theorem is referenced by: (None)