Theorem csbeq12 39293

Description: Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Assertion

Ref	Expression
csbeq12	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)

Proof of Theorem csbeq12

Step	Hyp	Ref	Expression
1		csbeq2 3856	. 2 ⊢ (∀𝑥 𝐶 = 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐷)
2		csbeq1 3854	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐵 / 𝑥⦌𝐷)
3	1, 2	sylan9eqr 2820	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ⦋csb 3851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743  df-csb 3852

This theorem is used by: (None)