Theorem dfss3f 3923

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 3922	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		df-ral 3078	. 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 2145  Ⅎwnfc 2908  ∀wral 3077   ⊆ wss 3899

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 2147  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2836  df-nfc 2910  df-ral 3078  df-ss 3916

This theorem is used by:  nfss  3924  nfchnd  18785  sigaclcu2  34752  bnj1498  35691  heibor1  38744  ssrabf  46128  ssrab2f  46131  limsupequzmpt2  46727  liminfequzmpt2  46800  pimconstlt1  47711  pimltpnff  47712  pimdecfgtioc  47724  pimincfltioc  47725  pimdecfgtioo  47726  pimincfltioo  47727  pimgtmnff  47731