Theorem dfss3f 3973

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Hypotheses

Ref	Expression
dfss2f.1	⊢ Ⅎ𝑥𝐴
dfss2f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
dfss3f	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Proof of Theorem dfss3f

Step	Hyp	Ref	Expression
1		dfss2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		dfss2f.2	. . 3 ⊢ Ⅎ𝑥𝐵
3	1, 2	dfss2f 3972	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 277	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2106  Ⅎwnfc 2883  ∀wral 3061   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-v 3476  df-in 3955  df-ss 3965

This theorem is referenced by:  nfss  3974  sigaclcu2  33113  bnj1498  34067  heibor1  36673  ssrabf  43793  ssrab2f  43796  limsupequzmpt2  44424  liminfequzmpt2  44497  pimconstlt1  45408  pimltpnff  45409  pimiooltgt  45416  pimdecfgtioc  45421  pimincfltioc  45422  pimdecfgtioo  45423  pimincfltioo  45424  pimgtmnff  45428  sssmf  45444