Theorem frege68c 40297

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege68c	⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))

Proof of Theorem frege68c

Step	Hyp	Ref	Expression
1		frege57aid 40238	. 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑))
2		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
3	2	frege67c 40296	. 2 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))
4	1, 3	ax-mp 5	1 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   ∈ wcel 2114  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793  ax-frege1 40156  ax-frege2 40157  ax-frege8 40175  ax-frege52a 40223  ax-frege58b 40267

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-tru 1540  df-fal 1550  df-ex 1781  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3773

This theorem is referenced by:  frege70  40299  frege77  40306  frege116  40345