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Theorem mof0ALT 47762
Description: Alternate proof for mof0 47760 with stronger requirements on distinct variables. Uses mo4 2554. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mof0ALT βˆƒ*𝑓 𝑓:π΄βŸΆβˆ…
Distinct variable group:   𝐴,𝑓

Proof of Theorem mof0ALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 f00 6766 . . . . 5 (𝑓:π΄βŸΆβˆ… ↔ (𝑓 = βˆ… ∧ 𝐴 = βˆ…))
21simplbi 497 . . . 4 (𝑓:π΄βŸΆβˆ… β†’ 𝑓 = βˆ…)
3 f00 6766 . . . . 5 (𝑔:π΄βŸΆβˆ… ↔ (𝑔 = βˆ… ∧ 𝐴 = βˆ…))
43simplbi 497 . . . 4 (𝑔:π΄βŸΆβˆ… β†’ 𝑔 = βˆ…)
5 eqtr3 2752 . . . 4 ((𝑓 = βˆ… ∧ 𝑔 = βˆ…) β†’ 𝑓 = 𝑔)
62, 4, 5syl2an 595 . . 3 ((𝑓:π΄βŸΆβˆ… ∧ 𝑔:π΄βŸΆβˆ…) β†’ 𝑓 = 𝑔)
76gen2 1790 . 2 βˆ€π‘“βˆ€π‘”((𝑓:π΄βŸΆβˆ… ∧ 𝑔:π΄βŸΆβˆ…) β†’ 𝑓 = 𝑔)
8 feq1 6691 . . 3 (𝑓 = 𝑔 β†’ (𝑓:π΄βŸΆβˆ… ↔ 𝑔:π΄βŸΆβˆ…))
98mo4 2554 . 2 (βˆƒ*𝑓 𝑓:π΄βŸΆβˆ… ↔ βˆ€π‘“βˆ€π‘”((𝑓:π΄βŸΆβˆ… ∧ 𝑔:π΄βŸΆβˆ…) β†’ 𝑓 = 𝑔))
107, 9mpbir 230 1 βˆƒ*𝑓 𝑓:π΄βŸΆβˆ…
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395  βˆ€wal 1531   = wceq 1533  βˆƒ*wmo 2526  βˆ…c0 4317  βŸΆwf 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6538  df-fn 6539  df-f 6540
This theorem is referenced by: (None)
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