Theorem csbeq12 34987

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 3816	. 2 ⊢ (∀𝑥 𝐶 = 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐷)
2		csbeq1 3814	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐵 / 𝑥⦌𝐷)
3	1, 2	sylan9eqr 2853	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520   = wceq 1522  ⦋csb 3811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-sbc 3707  df-csb 3812

This theorem is referenced by: (None)