Theorem sseq1d 2966

Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
sseq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
sseq1d	⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Proof of Theorem sseq1d

Step	Hyp	Ref	Expression
1		sseq1d.1	. 2 ⊢ (φ → A = B)
2		sseq1 2960	. 2 ⊢ (A = B → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))
3	1, 2	syl 14	1 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))

Syntax hints:   → wi 4   ↔ 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:  sseq12d  2968  eqsstrd  2973  snssg  3491  ssiun2s  3692  treq  3851  onsucsssucexmid  4212  funimass1  4919  feq1  4973  sbcfg  4988  fvmptssdm  5198  fvimacnvi  5224  nnsucsssuc  6010  ereq1  6049