Theorem dedths2 36906

Description: Generalization of dedths 36903 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 3713	. 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 4514	1 ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  [wsbc 3711  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sbc 3712  df-if 4457

This theorem is referenced by: (None)