Theorem dedth 3704

Description: Weak deduction theorem that eliminates a hypothesis φ, making it become an antecedent. We assume that a proof exists for φ when the class variable A is replaced with a specific class B. The hypothesis χ should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3711. If the inference has other hypotheses with class variable A, these can be kept by assigning keephyp 3717 to them. For more information, see the Deduction Theorem https://us.metamath.org/mpeuni/mmdeduction.html 3717. (Contributed by NM, 15-May-1999.)

Hypotheses

Ref	Expression
dedth.1	⊢ (A = if(φ, A, B) → (ψ ↔ χ))
dedth.2	⊢ χ

Assertion

Ref	Expression
dedth	⊢ (φ → ψ)

Proof of Theorem dedth

Step	Hyp	Ref	Expression
1		dedth.2	. 2 ⊢ χ
2		iftrue 3669	. . . 4 ⊢ (φ → if(φ, A, B) = A)
3	2	eqcomd 2358	. . 3 ⊢ (φ → A = if(φ, A, B))
4		dedth.1	. . 3 ⊢ (A = if(φ, A, B) → (ψ ↔ χ))
5	3, 4	syl 15	. 2 ⊢ (φ → (ψ ↔ χ))
6	1, 5	mpbiri 224	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  dedth2h  3705  dedth3h  3706