Theorem 3gencl 2715

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
3gencl.1	⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐷)
3gencl.2	⊢ (𝐹 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐹)
3gencl.3	⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧 ∈ 𝑅 𝐶 = 𝐺)
3gencl.4	⊢ (𝐴 = 𝐷 → (𝜑 ↔ 𝜓))
3gencl.5	⊢ (𝐵 = 𝐹 → (𝜓 ↔ 𝜒))
3gencl.6	⊢ (𝐶 = 𝐺 → (𝜒 ↔ 𝜃))
3gencl.7	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅) → 𝜑)

Assertion

Ref	Expression
3gencl	⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐷,𝑧   𝑧,𝐹   𝑥,𝑅,𝑦   𝑦,𝑆,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥)   𝑅(𝑧)   𝑆(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . . 5 ⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧 ∈ 𝑅 𝐶 = 𝐺)
2		df-rex 2420	. . . . 5 ⊢ (∃𝑧 ∈ 𝑅 𝐶 = 𝐺 ↔ ∃𝑧(𝑧 ∈ 𝑅 ∧ 𝐶 = 𝐺))
3	1, 2	bitri 183	. . . 4 ⊢ (𝐺 ∈ 𝑆 ↔ ∃𝑧(𝑧 ∈ 𝑅 ∧ 𝐶 = 𝐺))
4		3gencl.6	. . . . 5 ⊢ (𝐶 = 𝐺 → (𝜒 ↔ 𝜃))
5	4	imbi2d 229	. . . 4 ⊢ (𝐶 = 𝐺 → (((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → 𝜒) ↔ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → 𝜃)))
6		3gencl.1	. . . . . 6 ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐷)
7		3gencl.2	. . . . . 6 ⊢ (𝐹 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐹)
8		3gencl.4	. . . . . . 7 ⊢ (𝐴 = 𝐷 → (𝜑 ↔ 𝜓))
9	8	imbi2d 229	. . . . . 6 ⊢ (𝐴 = 𝐷 → ((𝑧 ∈ 𝑅 → 𝜑) ↔ (𝑧 ∈ 𝑅 → 𝜓)))
10		3gencl.5	. . . . . . 7 ⊢ (𝐵 = 𝐹 → (𝜓 ↔ 𝜒))
11	10	imbi2d 229	. . . . . 6 ⊢ (𝐵 = 𝐹 → ((𝑧 ∈ 𝑅 → 𝜓) ↔ (𝑧 ∈ 𝑅 → 𝜒)))
12		3gencl.7	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅) → 𝜑)
13	12	3expia 1183	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑧 ∈ 𝑅 → 𝜑))
14	6, 7, 9, 11, 13	2gencl 2714	. . . . 5 ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝑧 ∈ 𝑅 → 𝜒))
15	14	com12 30	. . . 4 ⊢ (𝑧 ∈ 𝑅 → ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → 𝜒))
16	3, 5, 15	gencl 2713	. . 3 ⊢ (𝐺 ∈ 𝑆 → ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → 𝜃))
17	16	com12 30	. 2 ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐺 ∈ 𝑆 → 𝜃))
18	17	3impia 1178	1 ⊢ ((𝐷 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-ie2 1470  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-3an 964  df-rex 2420

This theorem is referenced by:  axpre-ltwlin  7684  axpre-lttrn  7685  axpre-ltadd  7687  axpre-mulext  7689