Theorem dedths 39454

Description: A version of weak deduction theorem dedth 4513 using explicit substitution. (Contributed by NM, 15-Jun-2019.)

Hypothesis

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

Assertion

Ref	Expression
dedths	⊢ (𝜑 → 𝜓)

Proof of Theorem dedths

Step	Hyp	Ref	Expression
1		dfsbcq 3725	. . 3 ⊢ (𝑥 = if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) → ([𝑥 / 𝑥]𝜓 ↔ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓))
2		dedths.1	. . . 4 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓
3		sbcid 3740	. . . . 5 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
4		ifbi 4477	. . . . 5 ⊢ (([𝑥 / 𝑥]𝜑 ↔ 𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
5		dfsbcq 3725	. . . . 5 ⊢ (if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓)
7	2, 6	mpbir 232	. . 3 ⊢ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓
8	1, 7	dedth 4513	. 2 ⊢ ([𝑥 / 𝑥]𝜑 → [𝑥 / 𝑥]𝜓)
9		sbcid 3740	. 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓)
10	8, 3, 9	3imtr3i 292	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  [wsbc 3723  ifcif 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-sbc 3724  df-if 4455

This theorem is referenced by:  renegclALT  39455