Theorem frege120 43945

Description: Simplified application of one direction of dffrege115 43940. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege120	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Proof of Theorem frege120

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege120.a	. . . 4 ⊢ 𝐴 ∈ 𝑊
2	1	frege58c 43883	. . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋))
3		sbcim1 3861	. . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋))
4		sbcbr2g 5224	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎))
5	1, 4	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎)
6		csbvarg 4457	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴)
7	1, 6	ax-mp 5	. . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴
8	7	breq2i 5174	. . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴)
9	5, 8	bitri 275	. . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴)
10		sbceq1g 4440	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋))
11	1, 10	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋)
12	7	eqeq1i 2745	. . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
13	11, 12	bitri 275	. . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
14	3, 9, 13	3imtr3g 295	. . 3 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
15	2, 14	syl 17	. 2 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
16		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
17		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
18	16, 17	frege119 43944	. 2 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))
19	15, 18	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  [wsbc 3804  ⦋csb 3921   class class class wbr 5166  ^◡ccnv 5699  Fun wfun 6567

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-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771  ax-frege52a 43819  ax-frege58b 43863

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  frege121  43946