Theorem frege53b 44349

Description: Lemma for frege102 (via frege92 44414). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege53b	⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))

Proof of Theorem frege53b

Step	Hyp	Ref	Expression
1		frege52b 44348	. 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
2		ax-frege8 44268	. 2 ⊢ ((𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))

Syntax hints:   → wi 4  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-frege8 44268  ax-frege52c 44347

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-clab 2720  df-cleq 2733  df-clel 2816  df-sbc 3726

This theorem is referenced by:  frege55lem2b  44355