Theorem dedths2 39767

Description: Generalization of dedths 39764 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)

Hypothesis

Ref	Expression
dedths2.1	⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓

Assertion

Ref	Expression
dedths2	⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)

Proof of Theorem dedths2

Step	Hyp	Ref	Expression
1		dfsbcq 3745	. 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓))
2		dedths2.1	. 2 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓
3	1, 2	dedth 4545	1 ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)

This proof depends on syntax axioms:   → wi 4  [wsbc 3743  ifcif 4486

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3744  df-if 4487

This theorem is used by: (None)