Theorem sbcied2 3801

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 2787	. . 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 3800	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  [wsbc 3756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-sbc 3757

This theorem is referenced by:  iscat  17640  sectffval  17719  issubc  17804  isfunc  17833  cat1  18066  ismgm  18575  issgrp  18654  isnsg  19094  isrng  20070  isring  20153  isdomn  20621  islbs  20990  isassa  21772  opsrval  21960  isuhgr  28994  isushgr  28995  isupgr  29018  isumgr  29029  isuspgr  29086  isusgr  29087  isgrim  47886  isthinc  49412