Theorem csbeq12 35435

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 3887	. 2 ⊢ (∀𝑥 𝐶 = 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐷)
2		csbeq1 3885	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐵 / 𝑥⦌𝐷)
3	1, 2	sylan9eqr 2878	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531   = wceq 1533  ⦋csb 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772  df-csb 3883

This theorem is referenced by: (None)