Theorem sbcied2 3785

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbcied2.2	⊢ (𝜑 → 𝐴 = 𝐵)
sbcied2.3	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcied2	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied2

Step	Hyp	Ref	Expression
1		sbcied2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3		sbcied2.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	sylan9eqr 2793	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		sbcied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
6	4, 5	syldan 591	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	1, 6	sbcied 3784	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  [wsbc 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3741

This theorem is referenced by:  iscat  17595  sectffval  17674  issubc  17759  isfunc  17788  cat1  18021  ismgm  18566  issgrp  18645  isnsg  19084  isrng  20089  isring  20172  isdomn  20638  islbs  21028  isassa  21811  opsrval  22001  isuhgr  29133  isushgr  29134  isupgr  29157  isumgr  29168  isuspgr  29225  isusgr  29226  isgrim  48128  isthinc  49664