Theorem frege55lem2b 41366

Description: Lemma for frege55b 41367. 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 41364	. 2 ⊢ [𝑥 / 𝑧]𝑧 = 𝑥
2		frege53b 41360	. 2 ⊢ ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
3	1, 2	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Syntax hints:   → wi 4  [wsb 2072

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 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-13 2373  ax-ext 2710  ax-frege8 41279  ax-frege52c 41358

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-sbc 3713

This theorem is referenced by:  frege55b  41367