Theorem frege120

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

	Ref	Expression
Hypotheses	frege116.x	⊢ 𝑋 ∈ 𝑈
	frege118.y	⊢ 𝑌 ∈ 𝑉
	frege120.a	⊢ 𝐴 ∈ 𝑊
Assertion	frege120	⊢ ( Fun ◡ ◡ 𝑅 → ( 𝑌 𝑅 𝑋 → ( 𝑌 𝑅 𝐴 → 𝐴 = 𝑋 ) ) )

Proof

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