Theorem sbcied2 3852

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 2802	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		sbcied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
6	4, 5	syldan 590	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	1, 6	sbcied 3850	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805

This theorem is referenced by:  iscat  17730  sectffval  17811  issubc  17899  isfunc  17928  cat1  18164  ismgm  18679  issgrp  18758  isnsg  19195  isrng  20181  isring  20264  isdomn  20727  islbs  21098  isassa  21899  opsrval  22087  isuhgr  29095  isushgr  29096  isupgr  29119  isumgr  29130  isuspgr  29187  isusgr  29188  isgrim  47752  isthinc  48688