Theorem frege55lem2b 44739

Description: Lemma for frege55b 44740. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55lem2b	⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Proof of Theorem frege55lem2b

Step	Hyp	Ref	Expression
1		frege54cor1b 44737	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
2		frege53b 44733	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

This proof depends on syntax axioms:   → wi 4  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-13 2401  ax-ext 2732  ax-frege8 44652  ax-frege52c 44731

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3740

This theorem is used by:  frege55b  44740