Theorem csbeq12 38118

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 3926	. 2 ⊢ (∀𝑥 𝐶 = 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐷)
2		csbeq1 3924	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐵 / 𝑥⦌𝐷)
3	1, 2	sylan9eqr 2802	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ⦋csb 3921

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-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805  df-csb 3922

This theorem is referenced by: (None)