Theorem sbcied2 3763

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 2855	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		sbcied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
6	4, 5	syldan 594	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
7	1, 6	sbcied 3762	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721

This theorem is referenced by:  iscat  16935  sectffval  17012  issubc  17097  isfunc  17126  ismgm  17845  issgrp  17894  isnsg  18299  isring  19294  islbs  19841  isdomn  20060  isassa  20545  opsrval  20714  isuhgr  26853  isushgr  26854  isupgr  26877  isumgr  26888  isuspgr  26945  isusgr  26946  isrng  44500