Theorem 3sstr3d 2987

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3sstr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3sstr3d	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr3d

Step	Hyp	Ref	Expression
1		3sstr3d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr3d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3		3sstr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	sseq12d 2974	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷))
5	1, 4	mpbid 135	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by: (None)