Theorem frege68c 44677

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 44618	. 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑))
2		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
3	2	frege67c 44676	. 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 209  ∀wal 1568   ∈ wcel 2143  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-frege1 44536  ax-frege2 44537  ax-frege8 44555  ax-frege52a 44603  ax-frege58b 44647

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3745

This theorem is referenced by:  frege70  44679  frege77  44686  frege116  44725