Theorem dedths2 38346

Description: Generalization of dedths 38343 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 3774	. 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 4581	1 ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  [wsbc 3772  ifcif 4523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773  df-if 4524

This theorem is referenced by: (None)