Theorem 2gencl 3462

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

Hypotheses

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

Assertion

Ref	Expression
2gencl	⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑅   𝜓,𝑥   𝑦,𝐶   𝑦,𝑆   𝜒,𝑦

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

Proof of Theorem 2gencl

Step	Hyp	Ref	Expression
1		2gencl.2	. . . 4 ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷)
2		df-rex 3069	. . . 4 ⊢ (∃𝑦 ∈ 𝑅 𝐵 = 𝐷 ↔ ∃𝑦(𝑦 ∈ 𝑅 ∧ 𝐵 = 𝐷))
3	1, 2	bitri 274	. . 3 ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦(𝑦 ∈ 𝑅 ∧ 𝐵 = 𝐷))
4		2gencl.4	. . . 4 ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒))
5	4	imbi2d 340	. . 3 ⊢ (𝐵 = 𝐷 → ((𝐶 ∈ 𝑆 → 𝜓) ↔ (𝐶 ∈ 𝑆 → 𝜒)))
6		2gencl.1	. . . . . 6 ⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶)
7		df-rex 3069	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑅 𝐴 = 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝑅 ∧ 𝐴 = 𝐶))
8	6, 7	bitri 274	. . . . 5 ⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥(𝑥 ∈ 𝑅 ∧ 𝐴 = 𝐶))
9		2gencl.3	. . . . . 6 ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓))
10	9	imbi2d 340	. . . . 5 ⊢ (𝐴 = 𝐶 → ((𝑦 ∈ 𝑅 → 𝜑) ↔ (𝑦 ∈ 𝑅 → 𝜓)))
11		2gencl.5	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑)
12	11	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝑅 → (𝑦 ∈ 𝑅 → 𝜑))
13	8, 10, 12	gencl 3461	. . . 4 ⊢ (𝐶 ∈ 𝑆 → (𝑦 ∈ 𝑅 → 𝜓))
14	13	com12 32	. . 3 ⊢ (𝑦 ∈ 𝑅 → (𝐶 ∈ 𝑆 → 𝜓))
15	3, 5, 14	gencl 3461	. 2 ⊢ (𝐷 ∈ 𝑆 → (𝐶 ∈ 𝑆 → 𝜒))
16	15	impcom 407	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-rex 3069

This theorem is referenced by:  3gencl  3463  axaddrcl  10839  axmulrcl  10841  axpre-lttri  10852  axpre-mulgt0  10855  uzin2  14984