Theorem dfss3f 3914

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 3913	. 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 1540   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052   ⊆ wss 3890

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886  df-ral 3053  df-ss 3907

This theorem is referenced by:  nfss  3915  nfchnd  18566  sigaclcu2  34285  bnj1498  35224  heibor1  38142  ssrabf  45559  ssrab2f  45562  limsupequzmpt2  46161  liminfequzmpt2  46234  pimconstlt1  47145  pimltpnff  47146  pimdecfgtioc  47158  pimincfltioc  47159  pimdecfgtioo  47160  pimincfltioo  47161  pimgtmnff  47165  sssmf  47181