Theorem sseq1 2960

Description: Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
sseq1	⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Proof of Theorem sseq1

Step	Hyp	Ref	Expression
1		eqss 2954	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
2		sstr2 2946	. . . 4 ⊢ (B ⊆ A → (A ⊆ 𝐶 → B ⊆ 𝐶))
3	2	adantl 262	. . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ 𝐶 → B ⊆ 𝐶))
4		sstr2 2946	. . . 4 ⊢ (A ⊆ B → (B ⊆ 𝐶 → A ⊆ 𝐶))
5	4	adantr 261	. . 3 ⊢ ((A ⊆ B ∧ B ⊆ A) → (B ⊆ 𝐶 → A ⊆ 𝐶))
6	3, 5	impbid 120	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))
7	1, 6	sylbi 114	1 ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  sseq12  2962  sseq1i  2963  sseq1d  2966  nssne2  2996  psseq1  3025  sspsstr  3044  sbss  3323  pwjust  3352  elpw  3357  elpwg  3359  sssnr  3515  ssprr  3518  sstpr  3519  unimax  3605  trss  3854  elssabg  3893  bnd2  3917  mss  3953  exss  3954  ordtri2orexmid  4211  onsucsssucexmid  4212  sucprcreg  4227  tfis  4249  tfisi  4253  elnn  4271  nnregexmid  4285  releq  4365  xpsspw  4393  iss  4597  relcnvtr  4783  iotass  4827  fununi  4910  funcnvuni  4911  funimaexglem  4925  ffoss  5101  ssimaex  5177  tfrlem1  5864  nnsucsssuc  6010  qsss  6101  ssfiexmid  6254  elinp  6456  bj-om  9325  bj-2inf  9326  bj-nntrans  9339  bj-omtrans  9344