Theorem frege68c 44168

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 44109	. 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑))
2		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
3	2	frege67c 44167	. 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 206  ∀wal 1539   ∈ wcel 2113  [wsbc 3740

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-frege1 44027  ax-frege2 44028  ax-frege8 44046  ax-frege52a 44094  ax-frege58b 44138

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3741

This theorem is referenced by:  frege70  44170  frege77  44177  frege116  44216