Theorem dfss2f 3907

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 2372. (Revised by Gino Giotto, 19-May-2023.)

Hypotheses

Ref	Expression
dfss2f.1	⊢ Ⅎ𝑥𝐴
dfss2f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
dfss2f	⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Proof of Theorem dfss2f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3903	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵))
2		dfss2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2893	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		dfss2f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2893	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfim 1900	. . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)
7		nfv 1918	. . 3 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
8		eleq1w 2821	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9		eleq1w 2821	. . . 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 2340	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
12	1, 11	bitri 274	1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2108  Ⅎwnfc 2886   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  dfss3f  3908  ssrd  3922  ssrmof  3982  ss2ab  3989  rankval4  9556  rabexgfGS  30747  ballotth  32404  dvcosre  43343  itgsinexplem1  43385