Theorem dfss3f 3955

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

Hypotheses

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

Assertion

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

Proof of Theorem dfss3f

Step	Hyp	Ref	Expression
1		dfssf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		dfssf.2	. . 3 ⊢ Ⅎ𝑥𝐵
3	1, 2	dfssf 3954	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 278	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  Ⅎwnfc 2884  ∀wral 3052   ⊆ wss 3931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-clel 2810  df-nfc 2886  df-ral 3053  df-ss 3948

This theorem is referenced by:  nfss  3956  sigaclcu2  34156  bnj1498  35097  heibor1  37839  ssrabf  45105  ssrab2f  45108  limsupequzmpt2  45714  liminfequzmpt2  45787  pimconstlt1  46698  pimltpnff  46699  pimiooltgt  46706  pimdecfgtioc  46711  pimincfltioc  46712  pimdecfgtioo  46713  pimincfltioo  46714  pimgtmnff  46718  sssmf  46734