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 3080	. 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 1568   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-clel 2838  df-nfc 2912  df-ral 3080  df-ss 3923

This theorem is referenced by:  nfss  3931  nfchnd  18668  sigaclcu2  34491  bnj1498  35430  heibor1  38442  ssrabf  45815  ssrab2f  45818  limsupequzmpt2  46415  liminfequzmpt2  46488  pimconstlt1  47399  pimltpnff  47400  pimdecfgtioc  47412  pimincfltioc  47413  pimdecfgtioo  47414  pimincfltioo  47415  pimgtmnff  47419