Theorem dfssf 3999

Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) Avoid ax-13 2380. (Revised by GG, 19-May-2023.)

Hypotheses

Ref	Expression
dfssf.1	⊢ Ⅎ𝑥𝐴
dfssf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
dfssf	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Proof of Theorem dfssf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3993	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
2		dfssf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2900	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		dfssf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2900	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfim 1895	. . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)
7		nfv 1913	. . 3 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
8		eleq1w 2827	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9		eleq1w 2827	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
10	8, 9	imbi12d 344	. . 3 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
11	6, 7, 10	cbvalv1 2347	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
12	1, 11	bitri 275	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ wcel 2108  Ⅎwnfc 2893   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-clel 2819  df-nfc 2895  df-ss 3993

This theorem is referenced by:  dfss3f  4000  ssrd  4013  ssrmof  4076  ss2ab  4085  rankval4  9936  rabexgfGS  32527  ballotth  34502  dvcosre  45833  itgsinexplem1  45875