Theorem caovcang 5581

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypothesis

Ref	Expression
caovcang.1	⊢ ((φ ∧ (x ∈ 𝑇 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))

Assertion

Ref	Expression
caovcang	⊢ ((φ ∧ (A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   φ,x,y,z   x,𝐹,y,z   x,𝑆,y,z   x,𝑇,y,z

Proof of Theorem caovcang

Step	Hyp	Ref	Expression
1		caovcang.1	. . 3 ⊢ ((φ ∧ (x ∈ 𝑇 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y) = (x𝐹z) ↔ y = z))
2	1	ralrimivvva 2376	. 2 ⊢ (φ → ∀x ∈ 𝑇 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z))
3		oveq1 5439	. . . . 5 ⊢ (x = A → (x𝐹y) = (A𝐹y))
4		oveq1 5439	. . . . 5 ⊢ (x = A → (x𝐹z) = (A𝐹z))
5	3, 4	eqeq12d 2032	. . . 4 ⊢ (x = A → ((x𝐹y) = (x𝐹z) ↔ (A𝐹y) = (A𝐹z)))
6	5	bibi1d 222	. . 3 ⊢ (x = A → (((x𝐹y) = (x𝐹z) ↔ y = z) ↔ ((A𝐹y) = (A𝐹z) ↔ y = z)))
7		oveq2 5440	. . . . 5 ⊢ (y = B → (A𝐹y) = (A𝐹B))
8	7	eqeq1d 2026	. . . 4 ⊢ (y = B → ((A𝐹y) = (A𝐹z) ↔ (A𝐹B) = (A𝐹z)))
9		eqeq1 2024	. . . 4 ⊢ (y = B → (y = z ↔ B = z))
10	8, 9	bibi12d 224	. . 3 ⊢ (y = B → (((A𝐹y) = (A𝐹z) ↔ y = z) ↔ ((A𝐹B) = (A𝐹z) ↔ B = z)))
11		oveq2 5440	. . . . 5 ⊢ (z = 𝐶 → (A𝐹z) = (A𝐹𝐶))
12	11	eqeq2d 2029	. . . 4 ⊢ (z = 𝐶 → ((A𝐹B) = (A𝐹z) ↔ (A𝐹B) = (A𝐹𝐶)))
13		eqeq2 2027	. . . 4 ⊢ (z = 𝐶 → (B = z ↔ B = 𝐶))
14	12, 13	bibi12d 224	. . 3 ⊢ (z = 𝐶 → (((A𝐹B) = (A𝐹z) ↔ B = z) ↔ ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)))
15	6, 10, 14	rspc3v 2638	. 2 ⊢ ((A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀x ∈ 𝑇 ∀y ∈ 𝑆 ∀z ∈ 𝑆 ((x𝐹y) = (x𝐹z) ↔ y = z) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶)))
16	2, 15	mpan9 265	1 ⊢ ((φ ∧ (A ∈ 𝑇 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((A𝐹B) = (A𝐹𝐶) ↔ B = 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ∀wral 2280  (class class class)co 5432

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833  df-ov 5435

This theorem is referenced by:  caovcand  5582