Theorem dfss3f 3885

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 3884	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		df-ral 3075	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 281	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  Ⅎwnfc 2899  ∀wral 3070   ⊆ wss 3860

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-in 3867  df-ss 3877

This theorem is referenced by:  nfss  3886  sigaclcu2  31611  bnj1498  32565  heibor1  35554  ssrabf  42151  ssrab2f  42153  limsupequzmpt2  42754  liminfequzmpt2  42827  pimconstlt1  43734  pimltpnf  43735  pimiooltgt  43740  pimdecfgtioc  43744  pimincfltioc  43745  pimdecfgtioo  43746  pimincfltioo  43747  sssmf  43766