Theorem frege53b 43199

Description: Lemma for frege102 (via frege92 43264). 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 43198	. 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
2		ax-frege8 43118	. 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 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-frege8 43118  ax-frege52c 43197

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773

This theorem is referenced by:  frege55lem2b  43205