Theorem sseq12d 2968

Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Hypotheses

Ref	Expression
sseq1d.1	⊢ (φ → A = B)
sseq12d.2	⊢ (φ → 𝐶 = 𝐷)

Assertion

Ref	Expression
sseq12d	⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐷))

Proof of Theorem sseq12d

Step	Hyp	Ref	Expression
1		sseq1d.1	. . 3 ⊢ (φ → A = B)
2	1	sseq1d 2966	. 2 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐶))
3		sseq12d.2	. . 3 ⊢ (φ → 𝐶 = 𝐷)
4	3	sseq2d 2967	. 2 ⊢ (φ → (B ⊆ 𝐶 ↔ B ⊆ 𝐷))
5	2, 4	bitrd 177	1 ⊢ (φ → (A ⊆ 𝐶 ↔ B ⊆ 𝐷))

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:  3sstr3d  2981  3sstr4d  2982  ssdifeq0  3299  relcnvtr  4783  rdgisucinc  5912  nnaword  6020  nnawordi  6024