Theorem tposeqd 8170

Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypothesis

Ref	Expression
tposeqd.1	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
tposeqd	⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)

Proof of Theorem tposeqd

Step	Hyp	Ref	Expression
1		tposeqd.1	. 2 ⊢ (𝜑 → 𝐹 = 𝐺)
2		tposeq 8169	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
3	1, 2	syl 17	1 ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1542  tpos ctpos 8166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-tpos 8167

This theorem is referenced by:  oppcval  17637  oppchomfval  17638  oppccofval  17640  oppchomfpropd  17650  oppcmon  17663  oppgval  19280  oppgplusfval  19281  oppglsm  19575  opprval  20276  opprmulfval  20277  mattposvs  22398  mattpos1  22399  mamutpos  22401  mattposm  22402  madulid  22588  oppfvalg  49559  funcoppc4  49577  uptposlem  49630  oppgoppcco  50024