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Theorem sselOLD 3943
Description: Obsolete version of ssel 3942 as of 27-May-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sselOLD (𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Proof of Theorem sselOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3935 . . . . . 6 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
21biimpi 215 . . . . 5 (𝐴𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
3219.21bi 2183 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
43anim2d 613 . . 3 (𝐴𝐵 → ((𝑥 = 𝐶𝑥𝐴) → (𝑥 = 𝐶𝑥𝐵)))
54eximdv 1921 . 2 (𝐴𝐵 → (∃𝑥(𝑥 = 𝐶𝑥𝐴) → ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
6 dfclel 2816 . 2 (𝐶𝐴 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐴))
7 dfclel 2816 . 2 (𝐶𝐵 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵))
85, 6, 73imtr4g 296 1 (𝐴𝐵 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wss 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932
This theorem is referenced by: (None)
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