Theorem dfss3f 3930

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 3929	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		df-ral 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 281	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2146  Ⅎwnfc 2912  ∀wral 3081   ⊆ wss 3906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2840  df-nfc 2914  df-ral 3082  df-ss 3923

This theorem is used by:  nfss  3931  nfchnd  18684  sigaclcu2  34533  bnj1498  35473  heibor1  38494  ssrabf  45865  ssrab2f  45868  limsupequzmpt2  46465  liminfequzmpt2  46538  pimconstlt1  47449  pimltpnff  47450  pimdecfgtioc  47462  pimincfltioc  47463  pimdecfgtioo  47464  pimincfltioo  47465  pimgtmnff  47469